{"id":3719,"date":"2026-04-03T11:07:22","date_gmt":"2026-04-03T02:07:22","guid":{"rendered":"https:\/\/anagileway.com\/?p=3719"},"modified":"2026-04-21T10:43:05","modified_gmt":"2026-04-21T01:43:05","slug":"rank-of-flag-matrix","status":"publish","type":"post","link":"https:\/\/anagileway.com\/en\/2026\/04\/03\/rank-of-flag-matrix\/","title":{"rendered":"Flag Matrix Rank and Singular Value Decomposition"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">\u62d9\u8457<a href=\"https:\/\/www.kindaikagaku.co.jp\/book_list\/detail\/9784764961401\/\">\u300e\u56f3\u89e3\u3000\u7dda\u5f62\u4ee3\u6570\uff1a\u30b9\u30c8\u30e9\u30f3\u30b0\u6d41\u76f4\u611f\u7684\u7406\u89e3\u300f<\/a>\u306e\u30d6\u30ed\u30b0\u30b7\u30ea\u30fc\u30ba\u3067\u3059\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized has-lightbox\"><a href=\"https:\/\/www.kindaikagaku.co.jp\/book_list\/detail\/9784764906457\/\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"418\" height=\"592\" data-attachment-id=\"3689\" data-permalink=\"https:\/\/anagileway.com\/en\/2026\/04\/01\/rank-of-mondrian-matrix\/image-26\/\" data-orig-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image.png?fit=418%2C592&amp;ssl=1\" data-orig-size=\"418,592\" data-comments-opened=\"1\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image.png?fit=418%2C592&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image.png?resize=418%2C592&#038;ssl=1\" alt=\"\" class=\"wp-image-3689\" style=\"aspect-ratio:0.7061075147762454;width:174px;height:auto\" srcset=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image.png?w=418&amp;ssl=1 418w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image.png?resize=212%2C300&amp;ssl=1 212w\" sizes=\"auto, (max-width: 418px) 100vw, 418px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">\u524d\u56de\u3001<a href=\"https:\/\/anagileway.com\/2026\/04\/01\/rank-of-mondrian-matrix\/\">\u30e2\u30f3\u30c9\u30ea\u30a2\u30f3\u884c\u5217\u306e\u30e9\u30f3\u30af<\/a>\u3092\u8abf\u3079\u307e\u3057\u305f\u304c\u3001\u540c\u3058\u3088\u3046\u306a\u30eb\u30fc\u30eb\u3067\u5404\u56fd\u306e\u56fd\u65d7\u3092\u8abf\u3079\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u30e9\u30f3\u30af\u3068\u306f\u3001\u7dda\u5f62\u72ec\u7acb\u306a\u6700\u5927\u306a\u884c\u6570\u3067\u3059\u3002\u65d7\u306e\u4f8b\u3067\u306f\u3001\u8db3\u3057\u7b97\u3084\u8272\u306e\u4ea4\u63db\u3060\u3051\u3067\u91cd\u8907\u3092\u7701\u3044\u305f\u884c\u306e\u30d1\u30bf\u30fc\u30f3\u6570\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3061\u306a\u307f\u306b\u3001\u6b63\u78ba\u306b\u7dda\u5f62\u4ee3\u6570\u306e\u30e9\u30f3\u30af\u3092\u6e2c\u308b\u305f\u3081\u306e\u3001\u884c\u5217\u306e\u5909\u63db\u898f\u5247\u3092\u66f8\u3044\u3066\u304a\u304d\u307e\u3059\u3002\u4ee5\u4e0b\u306e\u884c\u57fa\u672c\u3092\u884c\u306a\u3063\u3066\u3082\u884c\u5217\u306e\u30e9\u30f3\u30af\u306f\u5909\u5316\u3057\u307e\u305b\u3093\u3002\uff08\uff09\u306e\u4e2d\u306b\u65d7\u3067\u8003\u3048\u308b\u3068\u304d\u306e\u308f\u304b\u308a\u3084\u3059\u3044\u89e3\u91c8\u3082\u5165\u308c\u307e\u3057\u305f\u306e\u3067\u3001\u3053\u306e\u5909\u63db\u3067\u7f6e\u304d\u63db\u3048\u3066\u304b\u3089\u3001\u6700\u5f8c\u306b\u6b8b\u308b\u30d1\u30bf\u30fc\u30f3\u306e\u6570\u3092\u6570\u3048\u3001\u81ea\u5206\u3067\u30e9\u30f3\u30af\u3092\u8a08\u7b97\u3057\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u884c\u57fa\u672c\u5909\u5f62<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u3042\u308b\u884c\u3092\u4ed6\u306e\u884c\u304b\u3089\u5f15\u304d\u7b97\u3059\u308b\uff08\u540c\u3058\u30d1\u30bf\u30fc\u30f3\u306e\u884c\u304c\u3042\u308c\u3070\u3001\u305d\u308c\u306f\u6d88\u305b\u308b\uff09\u3002<\/li>\n\n\n\n<li>\u3042\u308b\u884c\u306e\u03b1\u500d\u3092\u4ed6\u306e\u884c\u304b\u3089\u5f15\u304d\u7b97\u3059\u308b\uff08\u540c\u3058\u30d1\u30bf\u30fc\u30f3\u3067\u8272\u9055\u3044\u306e\u884c\u304c\u3042\u308c\u3070\u3001\u305d\u308c\u306f\u6d88\u305b\u308b\uff09\u3002<\/li>\n\n\n\n<li>\u3042\u308b\u884c\u3068\u3042\u308b\u884c\u3092\u4ea4\u63db\u3059\u308b\uff08\u8fd1\u3044\u30d1\u30bf\u30fc\u30f3\u306e\u884c\u306f\u8fd1\u304f\u306b\u79fb\u52d5\u3055\u305b\u3066\u8003\u3048\u3089\u308c\u308b\uff09\u3002<\/li>\n\n\n\n<li>\u3042\u308b\u884c\u3092\u975e\u96f6\u306e\u03b1\u500d\u3059\u308b\uff08\u3042\u308b\u884c\u306e\u8272\u3092\u5909\u3048\u308b\u3002\u767d\u306b\u306f\u5909\u3048\u3089\u308c\u306a\u3044\uff09\u3002<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u3055\u3089\u306b\u3001\u3053\u308c\u3089\u306e\u5909\u5f62\u3092\u884c\u3067\u306a\u304f\u5217\u306b\u3057\u3066\u3082\u5927\u4e08\u592b\u3002\u884c\u3068\u5217\u3092\u6df7\u305c\u3066\u5909\u5f62\u3057\u3066\u3082\u30e9\u30f3\u30af\u306f\u4e0d\u5909\u3067\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Fracne, Sweden, Greece<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/flag-rank.png?ssl=1\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"1000\" height=\"299\" data-attachment-id=\"3724\" data-permalink=\"https:\/\/anagileway.com\/en\/2026\/04\/03\/rank-of-flag-matrix\/flag-rank\/\" data-orig-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/flag-rank.png?fit=2340%2C700&amp;ssl=1\" data-orig-size=\"2340,700\" data-comments-opened=\"1\" data-image-title=\"flag-rank\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/flag-rank.png?fit=1000%2C299&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/flag-rank.png?resize=1000%2C299&#038;ssl=1\" alt=\"\" class=\"wp-image-3724\" srcset=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/flag-rank.png?resize=1024%2C306&amp;ssl=1 1024w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/flag-rank.png?resize=300%2C90&amp;ssl=1 300w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/flag-rank.png?resize=768%2C230&amp;ssl=1 768w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/flag-rank.png?resize=1536%2C459&amp;ssl=1 1536w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/flag-rank.png?resize=2048%2C613&amp;ssl=1 2048w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/flag-rank.png?resize=1200%2C359&amp;ssl=1 1200w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/flag-rank.png?resize=1440%2C431&amp;ssl=1 1440w\" sizes=\"auto, (max-width: 1000px) 100vw, 1000px\" \/><\/a><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">France \u306f\u3059\u3079\u3066\u304c\u540c\u3058\u884c\u306e\u7e70\u308a\u8fd4\u3057\u3067\u30d1\u30bf\u30fc\u30f3\u304c1\u3064\u3002Sweden \u306e\u884c\u306e\u30d1\u30bf\u30fc\u30f3\u306f2\u3064\u3067\u3059\u306d\u3002Greece\u306e\u884c\u306e\u30d1\u30bf\u30fc\u30f3\u306f\u5c11\u3057\u96e3\u3057\u3044\u3067\u3059\u304c\u30014\u3064\u3042\u308a\u305d\u3046\u3067\u3059\u3002\u3057\u304b\u3057\u30012\u884c\u76ee+3\u884c\u76ee=1\u884c\u76ee\u306a\u306e\u3067\u3001\u30e9\u30f3\u30af3\u3067\u3059\u3002\u3042\u308b\u3044\u306f\u3001\u5217\u306e\u30d1\u30bf\u30fc\u30f3\u6570\u3092\u6570\u3048\u305f\u65b9\u304c\u5206\u304b\u308a\u3084\u3059\u304f\u3001\u660e\u3089\u304b\u306b3\u30d1\u30bf\u30fc\u30f3\u3067\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Japan<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u3055\u3066\u3001\u554f\u984c\u306f\u65e5\u672c\u3067\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u884c\u5217\u306e\u554f\u984c\u3067\u306f\u300c\u659c\u3081\u306e\u30e9\u30a4\u30f3\u300d\u304c\u30e9\u30f3\u30af\u306b\u5927\u304d\u304f\u52b9\u304d\u307e\u3059\u3002\u884c\u57fa\u672c\u5909\u5f62\u3067\u6d88\u3048\u306b\u304f\u3044\u306e\u3067\u3059\u3002\u7e26\u3084\u6a2a\u306b\u3074\u3063\u305f\u308a\u6cbf\u3063\u305f\u30e9\u30a4\u30f3\uff08\u6c34\u5e73\u30fb\u5782\u76f4\uff09\u306f\u4ed6\u306e\u884c\u3084\u5217\u3068\u540c\u3058\u30d1\u30bf\u30fc\u30f3\u306b\u306a\u308b\u306e\u3067\u3001\u30e9\u30f3\u30af\u306e\u5897\u52a0\u306b\u52b9\u304d\u306b\u304f\u3044\u306e\u3067\u3059\u3002\u3053\u308c\u306b\u5bfe\u3057\u3066\u3001\u4e09\u89d2\u884c\u5217\u3084\u5358\u4f4d\u884c\u5217\u3092\u601d\u3044\u51fa\u3057\u3066\u4e0b\u3055\u3044\u3002\u659c\u3081\u306e\u30e9\u30a4\u30f3\u304c\u3042\u308b\u305f\u3081\u306b\u3001\u30e9\u30f3\u30af\u306f\u6700\u5927\uff08\u659c\u3081\u7dda\u306e\u30b5\u30a4\u30ba\uff09\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e0b\u56f3\u3092\u898b\u3066\u4e0b\u3055\u3044\u3002\u7e26\u6a2a\u306b\u6574\u5217\u3057\u305f\u6b63\u65b9\u5f62\u306f\u6700\u5c0f\u30e9\u30f3\u30af\u30671\u300145\u00b0\u50be\u3044\u305f\u6b63\u65b9\u5f62\uff1d\u30c0\u30a4\u30e4\u30e2\u30f3\u30c9\u306f\u6700\u5927\u30e9\u30f3\u30af\u3067$n$\u3002\u3068\u5927\u304d\u306a\u958b\u304d\u304c\u3042\u308a\u3001\u5186\u306f\u305d\u306e\u9593\u306e\u6570\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><a href=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-11.png?ssl=1\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"1000\" height=\"340\" data-attachment-id=\"3769\" data-permalink=\"https:\/\/anagileway.com\/en\/2026\/04\/03\/rank-of-flag-matrix\/image-37\/\" data-orig-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-11.png?fit=1606%2C546&amp;ssl=1\" data-orig-size=\"1606,546\" data-comments-opened=\"1\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-11.png?fit=1000%2C340&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-11.png?resize=1000%2C340&#038;ssl=1\" alt=\"\" class=\"wp-image-3769\" style=\"aspect-ratio:2.942662426169923;width:672px;height:auto\" srcset=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-11.png?resize=1024%2C348&amp;ssl=1 1024w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-11.png?resize=300%2C102&amp;ssl=1 300w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-11.png?resize=768%2C261&amp;ssl=1 768w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-11.png?resize=1536%2C522&amp;ssl=1 1536w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-11.png?resize=18%2C6&amp;ssl=1 18w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-11.png?resize=1200%2C408&amp;ssl=1 1200w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-11.png?resize=1440%2C490&amp;ssl=1 1440w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-11.png?w=1606&amp;ssl=1 1606w\" sizes=\"auto, (max-width: 1000px) 100vw, 1000px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Prof. Gilbert Strang \u3068 Prof. Alex Townsend \u306f\u3001\u3053\u306e\u5186\u306e\u30e9\u30f3\u30af\u3092 $0.5858n$ \u7a0b\u5ea6\u3068\u8a08\u7b97\u3057\u3066\u3044\u307e\u3059\u3002\u52d5\u753b\u300e\u306a\u305c\u7279\u7570\u5024\u306f\u6025\u6fc0\u306b\u6e1b\u5c11\u3059\u308b\u306e\u304b\u300f\u304a\u3088\u3073\u66f8\u7c4d\u300e\u30b9\u30c8\u30e9\u30f3\u30b0\uff1a\u6559\u990a\u306e\u7dda\u5f62\u4ee3\u6570\u300f\u306b\u306f\u3001\u65e5\u672c\u306e\u56fd\u65d7\u306e\u30e9\u30f3\u30af\u3092\u6c42\u3081\u308b\u554f\u984c\u304c\u3042\u308a\u307e\u3059\u3002\u89e3\u7b54\u306f\u52d5\u753b\u4e2d\u3067\u5206\u304b\u308a\u3084\u3059\u304f\u89e3\u8aac\u3055\u308c\u3066\u3044\u308b\u307b\u304b\u3001 <a href=\"https:\/\/www.kindaikagaku.co.jp\/pdf.php?sppid=269\">Solution Manual <\/a>\u306e 7.2 \u7bc0\u306e\u7df4\u7fd2\u554f\u984c 5\uff08p.82\uff09\u306b\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<div class=\"jetpack-video-wrapper\"><span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"1000\" height=\"563\" src=\"https:\/\/www.youtube.com\/embed\/9BYsNpTCZGg?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=en-US&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span><\/div>\n<\/div><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u3053\u3067\u3001\u89e3\u7b54\u306e\u3042\u3089\u3059\u3058\u3092\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u307e\u305a\u3001\u4e0b\u306e\u3088\u3046\u306b\u5186\u304b\u3089\u6b63\u65b9\u5f62\u3092\u304f\u308a\u629c\u304d\u307e\u3059\u3002\u3053\u306e\u6b63\u65b9\u5f62\u306f\u4e0a\u306b\u53d6\u308a\u51fa\u3057\u305f\u7d30\u9577\u3044\u9577\u65b9\u5f62\u306e rank1 \u884c\u5217\u3092\u5186\u304b\u3089\u5f15\u3044\u305f\u3068\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u7d30\u9577\u3044\u884c\u306f\u3001\u5186\u306e45\u00b0\u4f4d\u7f6e\u3068135\u00b0\u4f4d\u7f6e\u3092\u7d50\u3076\u76f4\u7dda\u3092\u53d6\u308a\u51fa\u3057\u305f\u3082\u306e\u3067\u3001\u30e9\u30f3\u30af\u306f1\u3067\u3001 $n$ \u304c\u5927\u304d\u304f\u306a\u308b\u3068\u7121\u8996\u3067\u304d\u307e\u3059\u3002<\/li>\n\n\n\n<li>\u6b21\u306b4\u3064\u306e\u5f13\u5f62\u304c\u5186\u306e\u5468\u56f2\u306b\u6b8b\u308a\u307e\u3059\u3002\u4e0a\u306e1\/4\u306f\u3001\u884c\u6570\u3068\u3057\u3066 $n &#8211; 1\/\\sqrt{2}n$ \u3042\u308a\u3001\u50be\u659c\u306e\u50be\u304d\u304c\u6700\u592745\u00b0\u3067\u3059\u3002\u3053\u306e\u90e8\u5206\u306f\u539a\u307f\u306e\u884c\u6570\u5206\u3060\u3051\u306e\u30e9\u30f3\u30af\u6570\u3068\u8003\u3048\u3089\u308c\u307e\u3059\u3002\u771f\u4e0b\u306e\u93e1\u50cf\u306e1\/4\u306f\u3053\u306e\u30b3\u30d4\u30fc\u3067\u3059\u306e\u3067\u3001\u30e9\u30f3\u30af\u3092\u5897\u3084\u3057\u307e\u305b\u3093\u3002<\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><a href=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-8.png?ssl=1\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"838\" height=\"592\" data-attachment-id=\"3738\" data-permalink=\"https:\/\/anagileway.com\/en\/2026\/04\/03\/rank-of-flag-matrix\/image-34\/\" data-orig-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-8.png?fit=838%2C592&amp;ssl=1\" data-orig-size=\"838,592\" data-comments-opened=\"1\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-8.png?fit=838%2C592&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-8.png?resize=838%2C592&#038;ssl=1\" alt=\"\" class=\"wp-image-3738\" style=\"aspect-ratio:1.415604558635107;width:328px;height:auto\" srcset=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-8.png?w=838&amp;ssl=1 838w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-8.png?resize=300%2C212&amp;ssl=1 300w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-8.png?resize=768%2C543&amp;ssl=1 768w\" sizes=\"auto, (max-width: 838px) 100vw, 838px\" \/><\/a><\/figure>\n<\/div>\n\n\n<ul class=\"wp-block-list\">\n<li>\u6b21\u306b\u5de6\u306e\u7e26\u306e\u5f13\u5f62\u3067\u3059\uff08\u53f3\u306e\u5f13\u5f62\u306f\u3053\u306e\u30b3\u30d4\u30fc\u3067\u7121\u8996\u3067\u304d\u307e\u3059\uff09\u3002\u3053\u306e\u90e8\u5206\u304c\u3059\u3079\u3066\u7dda\u5f62\u72ec\u7acb\u306a\uff08\u5225\u30d1\u30bf\u30fc\u30f3\u306e\uff09\u884c\u3068\u8003\u3048\u308c\u3070 $1\/\\sqrt{2}n$ \u306a\u306e\u3067\u3059\u304c\u3001\u5b9f\u969b\u306b\u306f\u4e2d\u592e\u306b\u8fd1\u3065\u304f\u306b\u3064\u308c\u5782\u76f4\u306b\u5207\u308a\u7acb\u3063\u3066\u3044\u304d\u3001\u540c\u3058\u884c\u3068\u307f\u306a\u305b\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\u305d\u3057\u3066\u3001\u3053\u306e\u7e26\u306e\u5f13\u5f62\u306e\u30e9\u30f3\u30af\u306f\u3001\u5148\u306b\u8a08\u7b97\u3057\u305f\u6a2a\u306e\u5f13\u5f62\u306e\u30e9\u30f3\u30af $n &#8211; 1\/\\sqrt{2}n$ \u3068\u540c\u3058\u3068\u8003\u3048\u3089\u308c\u307e\u3059\u3002\u306a\u305c\uff1f<\/li>\n\n\n\n<li>\u6a2a\u306e\u5f13\u5f62\u3067\u306f\u884c\u57fa\u672c\u5909\u5f62\u3067\u8003\u3048\u3066\u3044\u307e\u3057\u305f\u304c\u3001\u3053\u306e\u7e26\u306e\u5f13\u5f62\u3067\u306f\u5217\u57fa\u672c\u5909\u5f62\u3067\u8003\u3048\u308b\u306e\u3067\u3059\u3002\u5217\u3067\u8003\u3048\u308b\u3068\u305d\u306e\u50be\u659c\u304c\u5782\u76f4\u306b\u8fd1\u304f\u306a\u308b\u3053\u3068\u304c\u306a\u304f\u3001\u3059\u3079\u3066\u72ec\u7acb\u306a\u5217\u3068\u8003\u3048\u3089\u308c\u307e\u3059\u3002\u305d\u3057\u3066\u884c\u57fa\u672c\u5909\u5f62\u3068\u5217\u57fa\u672c\u5909\u5f62\u306f\u540c\u6642\u306b\u884c\u306a\u3063\u3066\u3082\u30e9\u30f3\u30af\u3092\u5909\u3048\u306a\u3044\u306e\u3067\u30011\u3064\u306e\u5f13\u5f62\u306e2\u500d\u306e\u30e9\u30f3\u30af\u304c\u5168\u4f53\u306e\u30e9\u30f3\u30af\u3068\u306a\u308a\u307e\u3059\u3002<\/li>\n\n\n\n<li>\u3088\u3063\u3066\u3001\u7d50\u8ad6\u3068\u3057\u3066\u3001\u65e5\u672c\u306e\u56fd\u65d7\u306e\u30e9\u30f3\u30af\u306f\u3001$n$ \u304c\u5927\u304d\u3044\u5834\u5408\u3001\u5c0f\u3055\u306a rank 1\u3092\u7121\u8996\u3057\u3066\u3001<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>2(n &#8211; 1\/\\sqrt{2}n) = (2 &#8211; \\sqrt{2})n \\approx 0.5858n<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u306a\u308a\u307e\u3059\u3002\u3067\u306f\u6570\u5024\u5b9f\u9a13\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u65e5\u672c\u56fd\u65d7\u306e\u30e9\u30f3\u30af\u8a08\u7b97<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u30ad\u30e3\u30f3\u30d0\u30b9\u30b5\u30a4\u30ba\u3092 200\u00d7300 \u30d4\u30af\u30bb\u30eb\u3001\u5186\u306e\u534a\u5f84\u3092 $n=100$ \u3068\u8a2d\u5b9a\u3057\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002\u30e9\u30f3\u30af\u306e\u8a08\u7b97\u306b\u306f\u7279\u7570\u5024\u5206\u89e3\u3092\u7528\u3044\u307e\u3059\uff08\u30e9\u30f3\u30af\u306f\u975e\u96f6\u7279\u7570\u5024\u306e\u6570\u3067\u3059\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7279\u7570\u5024\u5206\u89e3(compact) \u306f\u3001\u884c\u5217 $A \\in \\bR^{m \\times n}$ \u3092 $r$ \u3092\u30e9\u30f3\u30af\u3068\u3057\u3066\u30012\u3064\u306e\u76f4\u4ea4\u884c\u5217 $U \\in \\bR^{m \\times r}, V \\in \\bR^{n \\times r}$ \u3068\u5bfe\u89d2\u884c\u5217 $\\Sigma \\in \\bR^{r \\times r}$ \u306e\u7a4d\u306b\u5206\u89e3\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>A = U \\Sigma V\\transp = \\sigma_1 \\bu_1 \\bv_1\\transp +  \\sigma_2 \\bu_2 \\bv_2\\transp + \\cdots \\sigma_r \\bu_r \\bv_r\\transp <br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4f8b\u3048\u3070 $r=2$ \u306e\u5834\u5408\u3001\u3053\u3093\u306a\u98a8\u306b\u5206\u89e3\u3055\u308c\u3001\u7a4d\u3068\u548c\u306e\u4e21\u65b9\u3067\u8868\u73fe\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><a href=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-12.png?ssl=1\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"1000\" height=\"250\" data-attachment-id=\"3781\" data-permalink=\"https:\/\/anagileway.com\/en\/2026\/04\/03\/rank-of-flag-matrix\/image-38\/\" data-orig-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-12.png?fit=1192%2C298&amp;ssl=1\" data-orig-size=\"1192,298\" data-comments-opened=\"1\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-12.png?fit=1000%2C250&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-12.png?resize=1000%2C250&#038;ssl=1\" alt=\"\" class=\"wp-image-3781\" style=\"width:532px;height:auto\" srcset=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-12.png?resize=1024%2C256&amp;ssl=1 1024w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-12.png?resize=300%2C75&amp;ssl=1 300w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-12.png?resize=768%2C192&amp;ssl=1 768w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-12.png?resize=18%2C5&amp;ssl=1 18w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-12.png?w=1192&amp;ssl=1 1192w\" sizes=\"auto, (max-width: 1000px) 100vw, 1000px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"> $\\Sigma$ \u306e\u5bfe\u89d2\u306b\u306f\u3001\u7279\u7570\u5024\uff08$\\sigma_1 \\cdots \\sigma_n$ \uff09\u304c\u5358\u8abf\u6e1b\u5c11\u3067\u4e26\u3073\u307e\u3059\u3002\u305d\u3057\u3066\u3001\u975e\u96f6\u306e $\\sigma$ \u306e\u6570\u304c\u30e9\u30f3\u30af\u306b\u306a\u308a\u307e\u3059\u3002\u7279\u7570\u5024\u5206\u89e3\u3092\u4f7f\u3063\u3066\u7279\u7570\u5024\u3092\u53d6\u308a\u51fa\u3059\u3053\u3068\u3067\u3001\u5404\u30e9\u30f3\u30af\u306e\u300c\u5f37\u3055\u300d\u304c\u6b63\u306e\u5b9f\u6570\u3068\u3057\u3066\u53d6\u308a\u51fa\u305b\u308b\u306e\u3067\u3059\u3002$\\sigma_1$ \u306f\u6700\u5f37\u306e\u30b7\u30b0\u30ca\u30eb\u3001 $\\sigma_2$ \u306f\u305d\u306e\u6b21\u3001\u3068\u5f90\u3005\u306b\u30b7\u30b0\u30ca\u30eb\u304c\u5f31\u304f\u306a\u308a\u3001\u6700\u5f8c $\\sigma_{r+1}$ \u3067\u5b8c\u5168\u306b $0$ \u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\sigma_1 \\ge \\sigma_2 \\ge \\cdots \\ge \\sigma_r \\ge \\sigma_{r+1} = \\cdots = 0<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7279\u7570\u5024\u5206\u89e3\u306f\u305d\u306e\u5b89\u5b9a\u3057\u305f\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u3088\u3063\u3066\u3001\u7279\u7570\u5024\uff08\u5b9f\u6570\uff09\u306e\u5217\u3092\u4f5c\u308a\u3001\u305d\u306e\u975e\u96f6\u306e\u6570\uff08\u6b63\u306e\u6574\u6570\uff09\u304c\u30e9\u30f3\u30af\u3067\u3059\u3002\u30e9\u30f3\u30af\u306f\u6574\u6570\u3067\u975e\u96f6 $\\sigma$ \u7279\u7570\u5024\u306e\u6570\u3067\u3059\u304c\u3001\u96f6\u3068\u975e\u96f6\u306e\u5883\u754c\uff08 $\\epsilon$ \uff09\u3092\u3069\u3053\u306b\u304a\u304f\u304b\u3067\u3001\u73fe\u5b9f\u7684\u306b\u305d\u306e\u6570\u306f\u5909\u308f\u308a\u307e\u3059\u3002\u3053\u308c\u306b\u306f\u3001(1)\u305d\u3082\u305d\u3082\u884c\u5217\u306f\u6574\u6570\u306e\u884c\u6570\u3068\u5217\u6570\u3092\u6301\u3064\u306e\u3067\u5186\u3092\u7cbe\u5bc6\u306b\u306f\u8868\u73fe\u3067\u304d\u306a\u3044\u3053\u3068\uff08\u5b9f\u6570\u306e\u6574\u6570\u306b\u3088\u308b\u8fd1\u4f3c\uff09\u3001(2)\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u4e0a\u3067\u5b9f\u6570\u3092\u6d6e\u52d5\u5c0f\u6570\u70b9\u3067\u6271\u3046\u5834\u5408\u907f\u3051\u3089\u308c\u306a\u3044\u5fae\u5c11\u91cf\uff08\u6a5f\u68b0 $\\epsilon$ \uff1a $1 + \\epsilon = 1$ \u3068\u306a\u308b\u5024\uff09\u306e\u8907\u5408\u554f\u984c\u3067\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td><\/td><td>32bit (float)<\/td><td>64bit (double)<\/td><\/tr><tr><td>\u6a5f\u68b0 $\\epsilon$<\/td><td>$\\approx 10^{-7}$<\/td><td>$\\approx 10^{\u221216}$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u4eca\u56de\u8a2d\u5b9a\u3057\u305f\u534a\u5f84 $n=100$ \u3092\u3053\u306e\u516c\u5f0f\u306b\u5f53\u3066\u306f\u3081\u308b\u3068\u3001\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$(2-\\sqrt{2}) \\times 100 \\approx 0.58578 \\times 100 = 58.58$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3064\u307e\u308a\u3001\u7406\u8ad6\u4e0a\u306f\u300c58\u301c60\u500b\u3042\u305f\u308a\u306e\u7279\u7570\u5024\u3092\u7528\u610f\u3059\u308c\u3070\u5b8c\u74a7\u306a\u5186\u304c\u69cb\u6210\u3067\u304d\u3001\u305d\u308c\u4ee5\u964d\u306e\u7279\u7570\u5024\u306f\u3001\u8a08\u7b97\u6a5f\u4e0a\u8db3\u3057\u3066\u3082\u5909\u5316\u3057\u306a\u3044\u91cf\u306b\u843d\u3061\u8fbc\u3080\u306f\u305a\u3060\u300d\u3068\u3044\u3046\u4e88\u6e2c\u304c\u7acb\u3061\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u6570\u5024\u5b9f\u9a13\uff1a\u7279\u7570\u5024\u306e\u63a8\u79fb\u3068\u5fa9\u5143\u753b\u50cf<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u7406\u8ad6\u5024\u304c\u5b9f\u969b\u306e\u30c7\u30b8\u30bf\u30eb\u753b\u50cf\u3067\u3069\u306e\u7a0b\u5ea6\u6210\u308a\u7acb\u3064\u306e\u304b\u3001Python\u3092\u7528\u3044\u3066\u6570\u5024\u5b9f\u9a13\u3092\u884c\u3044\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u2460 \u7279\u7570\u5024\u306e\u30b0\u30e9\u30d5\uff08\u30bc\u30ed\u3078\u306e\u6025\u843d\uff09<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><a href=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/circle_rank_n100.png?ssl=1\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"1000\" height=\"600\" data-attachment-id=\"3750\" data-permalink=\"https:\/\/anagileway.com\/en\/2026\/04\/03\/rank-of-flag-matrix\/circle_rank_n100\/\" data-orig-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/circle_rank_n100.png?fit=1000%2C600&amp;ssl=1\" data-orig-size=\"1000,600\" data-comments-opened=\"1\" data-image-title=\"circle_rank_n100\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/circle_rank_n100.png?fit=1000%2C600&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/circle_rank_n100.png?resize=1000%2C600&#038;ssl=1\" alt=\"\" class=\"wp-image-3750\" style=\"aspect-ratio:1.6667402339439417;width:652px;height:auto\" srcset=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/circle_rank_n100.png?w=1000&amp;ssl=1 1000w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/circle_rank_n100.png?resize=300%2C180&amp;ssl=1 300w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/circle_rank_n100.png?resize=768%2C461&amp;ssl=1 768w\" sizes=\"auto, (max-width: 1000px) 100vw, 1000px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">\u8a08\u7b97\u3055\u308c\u305f\u7279\u7570\u5024\u3092\u5bfe\u6570\u30b0\u30e9\u30d5\u306b\u30d7\u30ed\u30c3\u30c8\u3057\u305f\u3068\u3053\u308d\u3001\u9a5a\u304f\u3079\u304d\u7d50\u679c\u3067\u3059\uff01\u7279\u7570\u5024\u306f58\u756a\u76ee\u4ed8\u8fd1\uff08\u7279\u7570\u5024\u306f\u5927\u304d\u3044\u65b9\u304b\u3089\u9806\u306b0\u304b\u3089\u756a\u53f7\u4ed8\u3051\u3055\u308c\u3066\u3044\u308b\uff09\u307e\u3067\u306f\u610f\u5473\u306e\u3042\u308b\u6570\u5024\u3092\u4fdd\u3063\u3066\u3044\u307e\u3057\u305f\u304c\u3001\u8d64\u3044\u70b9\u7dda\u3067\u793a\u3057\u305f <strong>\u7406\u8ad6\u5024 58.58<\/strong> \u306e\u30e9\u30a4\u30f3\u3092\u8d8a\u3048\u305f\u76f4\u5f8c\u3001$10^{-13}$ \u4ee5\u4e0b\u3078\u3068\u6025\u843d\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d7\u30ed\u30b0\u30e9\u30e0\u3067 $10^{-13}$ \u3088\u308a\u5927\u304d\u3044\u7279\u7570\u5024\u3092\u30ab\u30a6\u30f3\u30c8\u3059\u308b\u3068\u3001\u305d\u306e\u6570\u306f<strong>60\u500b<\/strong>\u3002\u8a55\u4fa1\u5f0f\u306e\u4e88\u6e2c\u3068\u6975\u3081\u3066\u9ad8\u3044\u7cbe\u5ea6\u3067\u4e00\u81f4\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e60\u3068\u3044\u3046\u6570\u3092\u3001\u6a5f\u68b0\u30e9\u30f3\u30af( Machine Rank )\u3084\u5b9f\u884c\u30e9\u30f3\u30af( Effective Rank )\u3001\u6570\u5024\u30e9\u30f3\u30af( Numerical Rank ) \u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Python\u306e mumpy \u304c\u7528\u3044\u308b\u6d6e\u52d5\u5c0f\u6570\u70b9\uff08np.float64\uff09\u306fC\u8a00\u8a9e\u306edouble\u306b\u5bfe\u5fdc\u3057\u3001\u6a5f\u68b0 $\\epsilon \\approx 10^{-16}$ \u3002\u3053\u306e\u5024\u306f1.0 \u4ed8\u8fd1\u306e\u8db3\u3057\u3066\u3082\u5909\u5316\u3057\u306a\u3044\u6700\u5c0f\u5024\u3068\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u5b9f\u969b\u306e\u8a08\u7b97\u3067\u306f\u8db3\u3059\u76f8\u624b\u306e\u6570\u306b\u3088\u3063\u3066\u8db3\u3057\u3066\u3082\u5909\u5316\u3057\u306a\u3044\u6700\u5c0f\u5024\u306f\u5909\u308f\u308a\u307e\u3059\u3002\u3053\u306e\u5b9f\u9a13\u3067\u306f $10^{-14}$ \u3042\u305f\u308a\u304c\u6700\u5c0f\u5024\u306b\u898b\u3048\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u2461 \u5fa9\u5143\u753b\u50cf\u306e\u63a8\u79fb\uff08n=1, 2, 3&#8230;\uff09<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>A = U \\Sigma V\\transp = \\sigma_1 \\bu_1 \\bv_1\\transp +  \\sigma_2 \\bu_2 \\bv_2\\transp + \\cdots<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u3044\u3046\u6e1b\u5c11\u3059\u308b\u7279\u7570\u5024\u5217\u304b\u3089\u5f97\u3089\u308c\u308b\u884c\u5217\u3092\u3001\u3042\u308b\u756a\u53f7\u3067\u6253\u3061\u5207\u308b\u3053\u3068\u3067\u300c\u8fd1\u4f3c\u300d\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\uff08\u3053\u308c\u304c\u6700\u826f\u8fd1\u4f3c\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u3092\u793a\u3059 Eckert-Young \u306e\u5b9a\u7406\u304c\u3042\u308a\u307e\u3059\uff09\u3002\u542b\u3081\u308b\u7279\u7570\u5024\u306e\u6570\u3092 $n=1, 2, 3&#8230;$ \u3068\u5f90\u3005\u306b\u5897\u3084\u3057\u3066\u3044\u304f\u3068\u3001\u753b\u50cf\u306f\u3069\u306e\u3088\u3046\u306b\u5909\u5316\u3059\u308b\u306e\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><a href=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-10.png?ssl=1\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"1000\" height=\"400\" data-attachment-id=\"3745\" data-permalink=\"https:\/\/anagileway.com\/en\/2026\/04\/03\/rank-of-flag-matrix\/image-36\/\" data-orig-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-10.png?fit=1600%2C640&amp;ssl=1\" data-orig-size=\"1600,640\" data-comments-opened=\"1\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-10.png?fit=1000%2C400&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-10.png?resize=1000%2C400&#038;ssl=1\" alt=\"\" class=\"wp-image-3745\" style=\"width:642px;height:auto\" srcset=\"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-10.png?resize=1024%2C410&amp;ssl=1 1024w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-10.png?resize=300%2C120&amp;ssl=1 300w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-10.png?resize=768%2C307&amp;ssl=1 768w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-10.png?resize=1536%2C614&amp;ssl=1 1536w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-10.png?resize=1200%2C480&amp;ssl=1 1200w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-10.png?resize=1440%2C576&amp;ssl=1 1440w, https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/image-10.png?w=1600&amp;ssl=1 1600w\" sizes=\"auto, (max-width: 1000px) 100vw, 1000px\" \/><\/a><\/figure>\n<\/div>\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Rank 1:<\/strong> \u5186\u306e\u9762\u5f71\u306f\u306a\u304f\u3001\u4e2d\u592e\u306b\u307c\u3093\u3084\u308a\u3068\u3057\u305f\u5341\u5b57\u578b\u306e\u5f71\u304c\u73fe\u308c\u307e\u3059\u3002<\/li>\n\n\n\n<li><strong>Rank 2:<\/strong> \u56db\u89d2\u5f62\u306e\u3088\u3046\u306a\u5f62\u304c\u6d6e\u304b\u3073\u4e0a\u304c\u308a\u307e\u3059\u3002\u60c5\u5831\u91cf\uff08\u30a8\u30cd\u30eb\u30ae\u30fc\uff09\u306e\u5927\u90e8\u5206\u306f\u3053\u3053\u3067\u65e2\u306b\u78ba\u4fdd\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/li>\n\n\n\n<li><strong>Rank 3\u301c4:<\/strong> \u3072\u3057\u5f62\u3084\u516b\u89d2\u5f62\u306e\u3088\u3046\u306a\u591a\u89d2\u5f62\u306b\u5909\u5316\u3057\u3001\u5f90\u3005\u306b\u300c\u4e38\u307f\u300d\u3092\u5e2f\u3073\u3066\u304d\u307e\u3059\u3002<\/li>\n\n\n\n<li><strong>Rank 10:<\/strong> \u30d1\u30c3\u3068\u898b\u306f\u300c\u5186\u300d\u3067\u3059\u304c\u3001\u5883\u754c\u90e8\u5206\u3092\u7121\u7406\u3084\u308a\u76f4\u4ea4\u6210\u5206\u3067\u4f5c\u3063\u3066\u3044\u308b\u305f\u3081\u3001\u6ce2\u7d0b\u72b6\u306e\u30ce\u30a4\u30ba\uff08\u30ea\u30f3\u30ae\u30f3\u30b0\uff09\u304c\u898b\u3048\u307e\u3059\u3002<\/li>\n\n\n\n<li><strong>Rank 50:<\/strong> \u307b\u307c\u5b8c\u74a7\u306a\u5186\u306b\u898b\u3048\u307e\u3059\u3002<\/li>\n\n\n\n<li><strong>Rank 60:<\/strong> \u7279\u7570\u5024\u304c\u30bc\u30ed\u306b\u843d\u3061\u8fbc\u3093\u3060\u5f8c\u306e\u753b\u50cf\u3067\u3059\u3002\u3053\u308c\u4ee5\u964d\u306f\u3044\u304f\u3089\u7279\u7570\u5024\u3092\u8db3\u3057\u3066\u3082\u3001\u753b\u50cf\u306f1\u30d4\u30af\u30bb\u30eb\u305f\u308a\u3068\u3082\u5909\u5316\u3057\u307e\u305b\u3093\uff08\u6a5f\u68b0 $\\epsilon $ \u4ed8\u8fd1\u306b\u5165\u3063\u305f\uff09\u3002\u3053\u308c\u4ee5\u4e0a\u306e\u5186\u306f\u30d4\u30af\u30bb\u30eb\u3067\u306f\u8868\u73fe\u3067\u304d\u307e\u305b\u3093\u3002<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">\u307e\u3068\u3081<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u753b\u50cf\u5727\u7e2e\u306e\u89b3\u70b9\u3067\u306f\u300c\u6700\u521d\u306e\u6570\u500b\uff08\u30e9\u30f3\u30af3\u301c4\uff09\u306e\u7279\u7570\u5024\u300d\u3060\u3051\u3067\u3001\u65e5\u306e\u4e38\u306e\u5927\u534a\u306e\u60c5\u5831\u306f\u8868\u73fe\u3067\u304d\u3066\u3057\u307e\u3044\u307e\u3059\u3002\u3057\u304b\u3057\u3001\u30d4\u30af\u30bb\u30eb\u306e\u4e16\u754c\u3067\u300c\u5b8c\u74a7\u306a\u4e38\u300d\u306e\u5883\u754c\u7dda\u3092\u7814\u304e\u6f84\u307e\u3059\u305f\u3081\u306b\u306f\u3001\u305d\u306e\u3042\u3068\u3082\u7d30\u304b\u306a\u7279\u7570\u5024\u304c\u30ed\u30f3\u30b0\u30c6\u30fc\u30eb\u3067\u4f7f\u308f\u308c\u7d9a\u3051\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u305d\u3057\u3066\u6700\u3082\u7f8e\u3057\u3044\u306e\u306f\u3001\u5b8c\u5168\u306a\u5186\u3092\u69cb\u6210\u3059\u308b\u305f\u3081\u306e\u300c\u6700\u5f8c\u306e\u30d4\u30fc\u30b9\u300d\u306e\u5834\u6240\u3067\u3059\u3002\u5186\u306e\u534a\u5f84\u306e <strong>\u7d0458.58%<\/strong> \u306e\u6570\u307e\u3067\u7279\u7570\u5024\u3092\u8db3\u3057\u5408\u308f\u305b\u308b\u3068\u3001\u30c7\u30b8\u30bf\u30eb\u4e0a\u3067\u8868\u73fe\u53ef\u80fd\u306a\u300c\u5b8c\u74a7\u306a\u5186\u300d\u306b\u5230\u9054\u3057\u3001\u305d\u308c\u4ee5\u4e0a\u306e\u60c5\u5831\u306f\u4e00\u5207\u4e0d\u8981\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$(2-\\sqrt{2})n \\approx 0.5858$ \u3068\u3044\u3046\u6570\u306f\u3001\u7dda\u5f62\u4ee3\u6570\uff08\u7279\u7570\u5024\uff09\u3001\u6570\u5024\u8a08\u7b97\uff08\u6d6e\u52d5\u5c0f\u6570\u306e\u9650\u754c\uff09\u3068\u753b\u50cf\u51e6\u7406\uff08\u6574\u6570\u500b\u306e\u30d4\u30af\u30bb\u30eb\uff09\u304c\u4ea4\u5dee\u3059\u308b\u70b9\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b7\u30f3\u30d7\u30eb\u306a\u56fd\u65d7\u306e\u56f3\u5f62\u306e\u4e2d\u306b\u3001\u3053\u306e\u3088\u3046\u306a\u6df1\u3044\u6570\u5b66\u306e\u7d50\u3073\u3064\u304d\u304c\u96a0\u3055\u308c\u3066\u3044\u308b\u3053\u3068\u306b\u611f\u52d5\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u8ffd\u8a18<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u8a18\u4e8b\u306e\u5143\u306b\u306a\u3063\u305f\u3001\u30e2\u30f3\u30c9\u30ea\u30a2\u30f3\u306e\u7d75\u753b\u306e\u3088\u3046\u306a\u30d6\u30ed\u30c3\u30af\u72b6\u884c\u5217\u3092\u3001\u7279\u7570\u5024\u5206\u89e3\u3067\u306a\u304f\u30ac\u30a6\u30b9\u306e\u6d88\u53bb\u6cd5\u3084 $CR$ \u5206\u89e3\u306b\u3088\u3063\u3066\u30e9\u30f3\u30af\u3092\u6c42\u3081\u308b\u8a18\u4e8b\u306b\u3064\u3044\u3066\u306f\u3001\u300c<a href=\"https:\/\/anagileway.com\/2026\/04\/01\/rank-of-mondrian-matrix\/\">\u30e2\u30f3\u30c9\u30ea\u30a2\u30f3\u884c\u5217\u306e\u30e9\u30f3\u30af<\/a>\u300d\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\uff08\u203bEdit  2026\/4\/4 : \u3053\u306e\u8a18\u4e8b\u3092 Prof. Gilbert Strang \u306b\u9001\u3063\u305f\u3068\u3053\u308d\u3001Prof. Alex Townsend \u3054\u672c\u4eba\u306b\u3082\u8a18\u4e8b\u3092\u7d39\u4ecb\u3057\u3066\u9802\u304d\u3001\u3055\u3063\u305d\u304f\u30df\u30b9\u3092\uff11\u3064\u6307\u6458\u3057\u3066\u9802\u3051\u307e\u3057\u305f\uff08\u30c0\u30a4\u30e4\u30e2\u30f3\u30c9\u306e\u30e9\u30f3\u30af\u306f $n$ \uff09\u3002\u307e\u305f\u3001\u3053\u306e\u5951\u6a5f\u306b\u3001WordPress \u306b MathJax Plugin \u3092\u5165\u308c\u3066\u3044\u3064\u3082\u4f7f\u3063\u3066\u3044\u308b TeX \u30b3\u30de\u30f3\u30c9\u3092\u30ab\u30b9\u30bf\u30de\u30a4\u30ba\u3057\u3001\u307e\u305f\u3001TranslatePress Pluigin \u3092\u5165\u308c\u3066\u82f1\u8a9e\u3078\u306e\u81ea\u52d5\u5909\u63db\u3092\u53f3\u4e0b\u306e\u30d5\u30ed\u30fc\u30c6\u30a3\u30f3\u30b0\u30dc\u30bf\u30f3\u304b\u3089\u53ef\u80fd\u306b\u3057\u3066\u3001\u82f1\u8a9e\u570f\u306e\u65b9\u306b\u53c2\u7167\u3057\u3084\u3059\u304f\u3057\u307e\u3057\u305f\u3002\uff09<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u4ed8\u9332\uff1a\u691c\u8a3c\u306b\u4f7f\u7528\u3057\u305fPython\u30b3\u30fc\u30c9<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u672c\u8a18\u4e8b\u306e\u30b0\u30e9\u30d5\u3068\u753b\u50cf\u306f\u3001\u4ee5\u4e0b\u306e\u30b3\u30fc\u30c9\u3067\u751f\u6210\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">\u7279\u7570\u5024\u5206\u89e3\u306b\u3088\u308b\u7279\u7570\u5024\u306e\u6e1b\u8870<\/h5>\n\n\n<div style=\"--line-number-gutter-width: 2ch\" class=\"show-line-numbers is-style-solarized-dark wp-block-code\">\n\t<div class=\"a8c\/code__header\"><div class=\"a8c\/code__header-right\"><button class=\"wp-element-button element-button a8c\/code__btn-copy\" type=\"button\" data-copy-text=\"import numpy as np\nimport matplotlib.pyplot as plt\n\n# \u8a2d\u5b9a\nheight = 200\nwidth = 300\nn = 100 # \u534a\u5f84\n\ny, x = np.ogrid[:height, :width]\ncy, cx = height \/ 2, width \/ 2\nmask = (x - cx)**2 + (y - cy)**2 &lt;= n**2\n\n# \u884c\u5217\u306e\u4f5c\u6210\uff08\u5186\u81ea\u4f53=1\u3001\u80cc\u666f=0\uff09\nA = np.zeros((height, width))\nA[mask] = 1\n\n# SVD\u306e\u8a08\u7b97\nU, S, Vt = np.linalg.svd(A, full_matrices=False)\n\n# \u4e88\u6e2c\u3055\u308c\u308b\u975e\u96f6\u7279\u7570\u5024\u306e\u6570\npredicted_rank = (2 - np.sqrt(2)) * n\n\nplt.figure(figsize=(10, 6))\n\n# \u7279\u7570\u5024\u306e\u30d7\u30ed\u30c3\u30c8\nplt.plot(S, marker=&#039;.&#039;, linestyle=&#039;-&#039;, color=&#039;b&#039;, label=&#039;Singular Values (Circle)&#039;)\n\n# (2-\u221a2)n \u306e\u30e9\u30a4\u30f3\nplt.axvline(x=predicted_rank, color=&#039;r&#039;, linestyle=&#039;--&#039;, linewidth=2, \n            label=f&#039;(2-\u221a2)n \u2248 {predicted_rank:.2f} (n={n})&#039;)\n\n# \u30bc\u30ed\u306e\u95be\u5024\u30e9\u30a4\u30f3\nplt.axhline(y=1e-10, color=&#039;g&#039;, linestyle=&#039;:&#039;, label=&#039;Zero Threshold (1e-10)&#039;)\n\nplt.yscale(&#039;log&#039;)\nplt.title(f&#039;Singular Values dropping to Zero (Circle Radius n={n}, Flag Size 200x300)&#039;)\nplt.xlabel(&#039;Index&#039;)\nplt.ylabel(&#039;Singular Value (Log Scale)&#039;)\nplt.xlim(0, 150) # n=100\u306a\u306e\u3067\u5c11\u3057\u5e83\u3081\u306b\u8868\u793a\nplt.ylim(1e-16, 1e4)\nplt.grid(True)\nplt.legend()\nplt.tight_layout()\nplt.savefig(&#039;circle_rank_n100.png&#039;)\n\n# \u5b9f\u969b\u306e\u975e\u96f6\u7279\u7570\u5024\u306e\u6570\u3092\u30ab\u30a6\u30f3\u30c8\nactual_rank = np.sum(S &gt; 1e-10)\nprint(f&quot;Radius (n): {n}&quot;)\nprint(f&quot;Predicted rank (2-\u221a2)n: {predicted_rank:.2f}&quot;)\nprint(f&quot;Actual non-zero singular values: {actual_rank}&quot;)\" hidden>Copy<\/button><span>Python<\/span><\/div><\/div>\n\t<div class=\"cm-editor\">\n\t\t<div class=\"cm-scroller\">\n\t\t\t\n<pre>\n<code class=\"language-python\"><div class=\"cm-line\"><span class=\"tok-keyword\">import<\/span> <span class=\"tok-variableName\">numpy<\/span> <span class=\"tok-keyword\">as<\/span> <span class=\"tok-variableName\">np<\/span><\/div><div class=\"cm-line\"><span class=\"tok-keyword\">import<\/span> <span class=\"tok-variableName\">matplotlib<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-variableName\">pyplot<\/span> <span class=\"tok-keyword\">as<\/span> <span class=\"tok-variableName\">plt<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># \u8a2d\u5b9a<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">height<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-number\">200<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">width<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-number\">300<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">n<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-number\">100<\/span> <span class=\"tok-comment\"># \u534a\u5f84<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">y<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">x<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">ogrid<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-variableName\">height<\/span><span class=\"tok-punctuation\">,<\/span> :<span class=\"tok-variableName\">width<\/span><span class=\"tok-punctuation\">]<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">cy<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">cx<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">height<\/span> <span class=\"tok-operator\">\/<\/span> <span class=\"tok-number\">2<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">width<\/span> <span class=\"tok-operator\">\/<\/span> <span class=\"tok-number\">2<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">mask<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">x<\/span> <span class=\"tok-operator\">-<\/span> <span class=\"tok-variableName\">cx<\/span><span class=\"tok-punctuation\">)<\/span><span class=\"tok-operator\">**<\/span><span class=\"tok-number\">2<\/span> <span class=\"tok-operator\">+<\/span> <span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">y<\/span> <span class=\"tok-operator\">-<\/span> <span class=\"tok-variableName\">cy<\/span><span class=\"tok-punctuation\">)<\/span><span class=\"tok-operator\">**<\/span><span class=\"tok-number\">2<\/span> <span class=\"tok-operator\">&lt;=<\/span> <span class=\"tok-variableName\">n<\/span><span class=\"tok-operator\">**<\/span><span class=\"tok-number\">2<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># \u884c\u5217\u306e\u4f5c\u6210\uff08\u5186\u81ea\u4f53=1\u3001\u80cc\u666f=0\uff09<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">A<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">zeros<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">height<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">width<\/span><span class=\"tok-punctuation\">)<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">A<\/span><span class=\"tok-punctuation\">[<\/span><span class=\"tok-variableName\">mask<\/span><span class=\"tok-punctuation\">]<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-number\">1<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># SVD\u306e\u8a08\u7b97<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">U<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">S<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">Vt<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">linalg<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">svd<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">A<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">full_matrices<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-bool\">False<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># \u4e88\u6e2c\u3055\u308c\u308b\u975e\u96f6\u7279\u7570\u5024\u306e\u6570<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">predicted_rank<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-punctuation\">(<\/span><span class=\"tok-number\">2<\/span> <span class=\"tok-operator\">-<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">sqrt<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-number\">2<\/span><span class=\"tok-punctuation\">)<\/span><span class=\"tok-punctuation\">)<\/span> <span class=\"tok-operator\">*<\/span> <span class=\"tok-variableName\">n<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">figure<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">figsize<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-number\">10<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">6<\/span><span class=\"tok-punctuation\">)<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># \u7279\u7570\u5024\u306e\u30d7\u30ed\u30c3\u30c8<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">plot<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">S<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">marker<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-string\">&apos;.&apos;<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">linestyle<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-string\">&apos;-&apos;<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">color<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-string\">&apos;b&apos;<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">label<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-string\">&apos;Singular Values (Circle)&apos;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># (2-\u221a2)n \u306e\u30e9\u30a4\u30f3<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">axvline<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">x<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-variableName\">predicted_rank<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">color<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-string\">&apos;r&apos;<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">linestyle<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-string\">&apos;--&apos;<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">linewidth<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-number\">2<\/span><span class=\"tok-punctuation\">,<\/span> <\/div><div class=\"cm-line\">            <span class=\"tok-variableName\">label<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-string2\">f&apos;(2-\u221a2)n \u2248 <\/span><span class=\"tok-punctuation\">{<\/span><span class=\"tok-variableName\">predicted_rank<\/span>:<span class=\"tok-keyword\">.2f<\/span><span class=\"tok-punctuation\">}<\/span><span class=\"tok-string2\"> (n=<\/span><span class=\"tok-punctuation\">{<\/span><span class=\"tok-variableName\">n<\/span><span class=\"tok-punctuation\">}<\/span><span class=\"tok-string2\">)&apos;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># \u30bc\u30ed\u306e\u95be\u5024\u30e9\u30a4\u30f3<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">axhline<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">y<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-number\">1e-10<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">color<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-string\">&apos;g&apos;<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">linestyle<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-string\">&apos;:&apos;<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">label<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-string\">&apos;Zero Threshold (1e-10)&apos;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">yscale<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-string\">&apos;log&apos;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">title<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-string2\">f&apos;Singular Values dropping to Zero (Circle Radius n=<\/span><span class=\"tok-punctuation\">{<\/span><span class=\"tok-variableName\">n<\/span><span class=\"tok-punctuation\">}<\/span><span class=\"tok-string2\">, Flag Size 200x300)&apos;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">xlabel<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-string\">&apos;Index&apos;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">ylabel<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-string\">&apos;Singular Value (Log Scale)&apos;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">xlim<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-number\">0<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">150<\/span><span class=\"tok-punctuation\">)<\/span> <span class=\"tok-comment\"># n=100\u306a\u306e\u3067\u5c11\u3057\u5e83\u3081\u306b\u8868\u793a<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">ylim<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-number\">1e-16<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">1e4<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">grid<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-bool\">True<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">legend<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">tight_layout<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">savefig<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-string\">&apos;circle_rank_n100.png&apos;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># \u5b9f\u969b\u306e\u975e\u96f6\u7279\u7570\u5024\u306e\u6570\u3092\u30ab\u30a6\u30f3\u30c8<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">actual_rank<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">sum<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">S<\/span> <span class=\"tok-operator\">&gt;<\/span> <span class=\"tok-number\">1e-10<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">print<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-string2\">f&quot;Radius (n): <\/span><span class=\"tok-punctuation\">{<\/span><span class=\"tok-variableName\">n<\/span><span class=\"tok-punctuation\">}<\/span><span class=\"tok-string2\">&quot;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">print<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-string2\">f&quot;Predicted rank (2-\u221a2)n: <\/span><span class=\"tok-punctuation\">{<\/span><span class=\"tok-variableName\">predicted_rank<\/span>:<span class=\"tok-keyword\">.2f<\/span><span class=\"tok-punctuation\">}<\/span><span class=\"tok-string2\">&quot;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">print<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-string2\">f&quot;Actual non-zero singular values: <\/span><span class=\"tok-punctuation\">{<\/span><span class=\"tok-variableName\">actual_rank<\/span><span class=\"tok-punctuation\">}<\/span><span class=\"tok-string2\">&quot;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><\/code><\/pre>\n\t\t<\/div>\n\t<\/div>\n<\/div>\n\n\n<h5 class=\"wp-block-heading\">\u56fd\u65d7\u753b\u50cf\u306e\u5fa9\u5143<\/h5>\n\n\n<div style=\"--line-number-gutter-width: 2ch\" class=\"show-line-numbers is-style-solarized-dark wp-block-code\">\n\t<div class=\"a8c\/code__header\"><div class=\"a8c\/code__header-right\"><button class=\"wp-element-button element-button a8c\/code__btn-copy\" type=\"button\" data-copy-text=\"import numpy as np\nimport matplotlib.pyplot as plt\n\n# \u30ad\u30e3\u30f3\u30d0\u30b9\u30b5\u30a4\u30ba\u3068\u5186\u306e\u534a\u5f84\u306e\u8a2d\u5b9a\nheight = 200\nwidth = 300\nr = 50\n\n# RGB\u306e\u753b\u50cf\u914d\u5217\u3092\u4f5c\u6210\uff08\u80cc\u666f\u306f\u767d: 1.0, 1.0, 1.0\uff09\nimg = np.ones((height, width, 3))\n\n# \u65e5\u306e\u4e38\u306e\u8d64\u8272\uff08#BC002D -&gt; RGB: 188\/255, 0, 45\/255\uff09\nred_color = np.array([188\/255, 0, 45\/255])\n\n# \u5186\u5f62\u30de\u30b9\u30af\u306e\u4f5c\u6210\ny, x = np.ogrid[:height, :width]\ncy, cx = height \/ 2, width \/ 2\nmask = (x - cx)**2 + (y - cy)**2 &lt;= r**2\n\n# \u30de\u30b9\u30af\u90e8\u5206\u3092\u8d64\u8272\u306b\nimg[mask] = red_color\n\n# RGB\u5404\u30c1\u30e3\u30f3\u30cd\u30eb\u3054\u3068\u306b\u7279\u7570\u5024\u5206\u89e3 (SVD) \u3092\u5b9f\u884c\nU_r, S_r, Vt_r = np.linalg.svd(img[:,:,0], full_matrices=False)\nU_g, S_g, Vt_g = np.linalg.svd(img[:,:,1], full_matrices=False)\nU_b, S_b, Vt_b = np.linalg.svd(img[:,:,2], full_matrices=False)\n\n# \u6307\u5b9a\u3055\u308c\u305f\u30e9\u30f3\u30af\nranks = [1, 2, 3, 4, 5, 10, 20, 30, 40, 50]\n\nfig, axes = plt.subplots(2, 5, figsize=(15, 6))\naxes = axes.flatten()\n\nfor i, k in enumerate(ranks):\n    # \u5404\u30c1\u30e3\u30f3\u30cd\u30eb\u3054\u3068\u306b\u30e9\u30f3\u30afk\u3067\u753b\u50cf\u3092\u518d\u69cb\u6210\n    R_k = U_r[:, :k] @ np.diag(S_r[:k]) @ Vt_r[:k, :]\n    G_k = U_g[:, :k] @ np.diag(S_g[:k]) @ Vt_g[:k, :]\n    B_k = U_b[:, :k] @ np.diag(S_b[:k]) @ Vt_b[:k, :]\n    \n    # \u30c1\u30e3\u30f3\u30cd\u30eb\u3092\u7d50\u5408\u3057\u3066RGB\u753b\u50cf\u306b\uff08\u5024\u304c0\u672a\u6e80\u30841\u3092\u8d85\u904e\u3059\u308b\u5834\u5408\u306f\u30af\u30ea\u30c3\u30d7\uff09\n    img_k = np.stack((R_k, G_k, B_k), axis=2)\n    img_k = np.clip(img_k, 0, 1)\n    \n    # \u30d7\u30ed\u30c3\u30c8\n    axes[i].imshow(img_k)\n    axes[i].set_title(f&#039;Rank n={k}&#039;)\n    axes[i].axis(&#039;off&#039;)\n    \n    # \u308f\u304b\u308a\u3084\u3059\u3044\u3088\u3046\u306b\u8584\u3044\u67a0\u7dda\u3092\u8ffd\u52a0\n    rect = plt.Rectangle((0, 0), width-1, height-1, fill=False, edgecolor=&#039;gray&#039;, linewidth=0.5)\n    axes[i].add_patch(rect)\n\nplt.tight_layout()\nplt.savefig(&#039;svd_color_flag_reconstruction.png&#039;, dpi=150)\" hidden>Copy<\/button><span>Python<\/span><\/div><\/div>\n\t<div class=\"cm-editor\">\n\t\t<div class=\"cm-scroller\">\n\t\t\t\n<pre>\n<code class=\"language-python\"><div class=\"cm-line\"><span class=\"tok-keyword\">import<\/span> <span class=\"tok-variableName\">numpy<\/span> <span class=\"tok-keyword\">as<\/span> <span class=\"tok-variableName\">np<\/span><\/div><div class=\"cm-line\"><span class=\"tok-keyword\">import<\/span> <span class=\"tok-variableName\">matplotlib<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-variableName\">pyplot<\/span> <span class=\"tok-keyword\">as<\/span> <span class=\"tok-variableName\">plt<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># \u30ad\u30e3\u30f3\u30d0\u30b9\u30b5\u30a4\u30ba\u3068\u5186\u306e\u534a\u5f84\u306e\u8a2d\u5b9a<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">height<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-number\">200<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">width<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-number\">300<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">r<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-number\">50<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># RGB\u306e\u753b\u50cf\u914d\u5217\u3092\u4f5c\u6210\uff08\u80cc\u666f\u306f\u767d: 1.0, 1.0, 1.0\uff09<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">img<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">ones<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">height<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">width<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">3<\/span><span class=\"tok-punctuation\">)<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># \u65e5\u306e\u4e38\u306e\u8d64\u8272\uff08#BC002D -&gt; RGB: 188\/255, 0, 45\/255\uff09<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">red_color<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">array<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-punctuation\">[<\/span><span class=\"tok-number\">188<\/span><span class=\"tok-operator\">\/<\/span><span class=\"tok-number\">255<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">0<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">45<\/span><span class=\"tok-operator\">\/<\/span><span class=\"tok-number\">255<\/span><span class=\"tok-punctuation\">]<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># \u5186\u5f62\u30de\u30b9\u30af\u306e\u4f5c\u6210<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">y<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">x<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">ogrid<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-variableName\">height<\/span><span class=\"tok-punctuation\">,<\/span> :<span class=\"tok-variableName\">width<\/span><span class=\"tok-punctuation\">]<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">cy<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">cx<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">height<\/span> <span class=\"tok-operator\">\/<\/span> <span class=\"tok-number\">2<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">width<\/span> <span class=\"tok-operator\">\/<\/span> <span class=\"tok-number\">2<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">mask<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">x<\/span> <span class=\"tok-operator\">-<\/span> <span class=\"tok-variableName\">cx<\/span><span class=\"tok-punctuation\">)<\/span><span class=\"tok-operator\">**<\/span><span class=\"tok-number\">2<\/span> <span class=\"tok-operator\">+<\/span> <span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">y<\/span> <span class=\"tok-operator\">-<\/span> <span class=\"tok-variableName\">cy<\/span><span class=\"tok-punctuation\">)<\/span><span class=\"tok-operator\">**<\/span><span class=\"tok-number\">2<\/span> <span class=\"tok-operator\">&lt;=<\/span> <span class=\"tok-variableName\">r<\/span><span class=\"tok-operator\">**<\/span><span class=\"tok-number\">2<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># \u30de\u30b9\u30af\u90e8\u5206\u3092\u8d64\u8272\u306b<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">img<\/span><span class=\"tok-punctuation\">[<\/span><span class=\"tok-variableName\">mask<\/span><span class=\"tok-punctuation\">]<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">red_color<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># RGB\u5404\u30c1\u30e3\u30f3\u30cd\u30eb\u3054\u3068\u306b\u7279\u7570\u5024\u5206\u89e3 (SVD) \u3092\u5b9f\u884c<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">U_r<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">S_r<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">Vt_r<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">linalg<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">svd<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">img<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-punctuation\">,<\/span>:<span class=\"tok-punctuation\">,<\/span><span class=\"tok-number\">0<\/span><span class=\"tok-punctuation\">]<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">full_matrices<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-bool\">False<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">U_g<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">S_g<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">Vt_g<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">linalg<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">svd<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">img<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-punctuation\">,<\/span>:<span class=\"tok-punctuation\">,<\/span><span class=\"tok-number\">1<\/span><span class=\"tok-punctuation\">]<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">full_matrices<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-bool\">False<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">U_b<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">S_b<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">Vt_b<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">linalg<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">svd<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">img<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-punctuation\">,<\/span>:<span class=\"tok-punctuation\">,<\/span><span class=\"tok-number\">2<\/span><span class=\"tok-punctuation\">]<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">full_matrices<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-bool\">False<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-comment\"># \u6307\u5b9a\u3055\u308c\u305f\u30e9\u30f3\u30af<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">ranks<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-punctuation\">[<\/span><span class=\"tok-number\">1<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">2<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">3<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">4<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">5<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">10<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">20<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">30<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">40<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">50<\/span><span class=\"tok-punctuation\">]<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">fig<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">axes<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">subplots<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-number\">2<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">5<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">figsize<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-number\">15<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">6<\/span><span class=\"tok-punctuation\">)<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">axes<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">axes<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">flatten<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-keyword\">for<\/span> <span class=\"tok-variableName\">i<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">k<\/span> <span class=\"tok-keyword\">in<\/span> <span class=\"tok-variableName\">enumerate<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">ranks<\/span><span class=\"tok-punctuation\">)<\/span>:<\/div><div class=\"cm-line\">    <span class=\"tok-comment\"># \u5404\u30c1\u30e3\u30f3\u30cd\u30eb\u3054\u3068\u306b\u30e9\u30f3\u30afk\u3067\u753b\u50cf\u3092\u518d\u69cb\u6210<\/span><\/div><div class=\"cm-line\">    <span class=\"tok-variableName\">R_k<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">U_r<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-punctuation\">,<\/span> :<span class=\"tok-variableName\">k<\/span><span class=\"tok-punctuation\">]<\/span> <span class=\"tok-operator\">@<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">diag<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">S_r<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-variableName\">k<\/span><span class=\"tok-punctuation\">]<\/span><span class=\"tok-punctuation\">)<\/span> <span class=\"tok-operator\">@<\/span> <span class=\"tok-variableName\">Vt_r<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-variableName\">k<\/span><span class=\"tok-punctuation\">,<\/span> :<span class=\"tok-punctuation\">]<\/span><\/div><div class=\"cm-line\">    <span class=\"tok-variableName\">G_k<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">U_g<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-punctuation\">,<\/span> :<span class=\"tok-variableName\">k<\/span><span class=\"tok-punctuation\">]<\/span> <span class=\"tok-operator\">@<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">diag<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">S_g<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-variableName\">k<\/span><span class=\"tok-punctuation\">]<\/span><span class=\"tok-punctuation\">)<\/span> <span class=\"tok-operator\">@<\/span> <span class=\"tok-variableName\">Vt_g<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-variableName\">k<\/span><span class=\"tok-punctuation\">,<\/span> :<span class=\"tok-punctuation\">]<\/span><\/div><div class=\"cm-line\">    <span class=\"tok-variableName\">B_k<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">U_b<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-punctuation\">,<\/span> :<span class=\"tok-variableName\">k<\/span><span class=\"tok-punctuation\">]<\/span> <span class=\"tok-operator\">@<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">diag<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">S_b<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-variableName\">k<\/span><span class=\"tok-punctuation\">]<\/span><span class=\"tok-punctuation\">)<\/span> <span class=\"tok-operator\">@<\/span> <span class=\"tok-variableName\">Vt_b<\/span><span class=\"tok-punctuation\">[<\/span>:<span class=\"tok-variableName\">k<\/span><span class=\"tok-punctuation\">,<\/span> :<span class=\"tok-punctuation\">]<\/span><\/div><div class=\"cm-line\">    <\/div><div class=\"cm-line\">    <span class=\"tok-comment\"># \u30c1\u30e3\u30f3\u30cd\u30eb\u3092\u7d50\u5408\u3057\u3066RGB\u753b\u50cf\u306b\uff08\u5024\u304c0\u672a\u6e80\u30841\u3092\u8d85\u904e\u3059\u308b\u5834\u5408\u306f\u30af\u30ea\u30c3\u30d7\uff09<\/span><\/div><div class=\"cm-line\">    <span class=\"tok-variableName\">img_k<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">stack<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">R_k<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">G_k<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">B_k<\/span><span class=\"tok-punctuation\">)<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">axis<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-number\">2<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\">    <span class=\"tok-variableName\">img_k<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">np<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">clip<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">img_k<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">0<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">1<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\">    <\/div><div class=\"cm-line\">    <span class=\"tok-comment\"># \u30d7\u30ed\u30c3\u30c8<\/span><\/div><div class=\"cm-line\">    <span class=\"tok-variableName\">axes<\/span><span class=\"tok-punctuation\">[<\/span><span class=\"tok-variableName\">i<\/span><span class=\"tok-punctuation\">]<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">imshow<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">img_k<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\">    <span class=\"tok-variableName\">axes<\/span><span class=\"tok-punctuation\">[<\/span><span class=\"tok-variableName\">i<\/span><span class=\"tok-punctuation\">]<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">set_title<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-string2\">f&apos;Rank n=<\/span><span class=\"tok-punctuation\">{<\/span><span class=\"tok-variableName\">k<\/span><span class=\"tok-punctuation\">}<\/span><span class=\"tok-string2\">&apos;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\">    <span class=\"tok-variableName\">axes<\/span><span class=\"tok-punctuation\">[<\/span><span class=\"tok-variableName\">i<\/span><span class=\"tok-punctuation\">]<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">axis<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-string\">&apos;off&apos;<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\">    <\/div><div class=\"cm-line\">    <span class=\"tok-comment\"># \u308f\u304b\u308a\u3084\u3059\u3044\u3088\u3046\u306b\u8584\u3044\u67a0\u7dda\u3092\u8ffd\u52a0<\/span><\/div><div class=\"cm-line\">    <span class=\"tok-variableName\">rect<\/span> <span class=\"tok-operator\">=<\/span> <span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">Rectangle<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-number\">0<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-number\">0<\/span><span class=\"tok-punctuation\">)<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">width<\/span><span class=\"tok-operator\">-<\/span><span class=\"tok-number\">1<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">height<\/span><span class=\"tok-operator\">-<\/span><span class=\"tok-number\">1<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">fill<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-bool\">False<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">edgecolor<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-string\">&apos;gray&apos;<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">linewidth<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-number\">0.5<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\">    <span class=\"tok-variableName\">axes<\/span><span class=\"tok-punctuation\">[<\/span><span class=\"tok-variableName\">i<\/span><span class=\"tok-punctuation\">]<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">add_patch<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-variableName\">rect<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">tight_layout<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-punctuation\">)<\/span><\/div><div class=\"cm-line\"><span class=\"tok-variableName\">plt<\/span><span class=\"tok-operator\">.<\/span><span class=\"tok-propertyName\">savefig<\/span><span class=\"tok-punctuation\">(<\/span><span class=\"tok-string\">&apos;svd_color_flag_reconstruction.png&apos;<\/span><span class=\"tok-punctuation\">,<\/span> <span class=\"tok-variableName\">dpi<\/span><span class=\"tok-operator\">=<\/span><span class=\"tok-number\">150<\/span><span class=\"tok-punctuation\">)<\/span><\/div><\/code><\/pre>\n\t\t<\/div>\n\t<\/div>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u62d9\u8457\u300e\u56f3\u89e3\u3000\u7dda\u5f62\u4ee3\u6570\uff1a\u30b9\u30c8\u30e9\u30f3\u30b0\u6d41\u76f4\u611f\u7684\u7406\u89e3\u300f\u306e\u30d6\u30ed\u30b0\u30b7\u30ea\u30fc\u30ba\u3067\u3059\u3002 \u524d\u56de\u3001\u30e2\u30f3\u30c9\u30ea\u30a2\u30f3\u884c\u5217\u306e\u30e9\u30f3\u30af\u3092\u8abf\u3079\u307e\u3057 [&hellip;]<\/p>\n","protected":false},"author":24727826,"featured_media":3724,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"jetpack_seo_schema_type":"","_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_wpcom_ai_launchpad_first_post":false,"_jetpack_hide_featured_image":false,"_jetpack_featured_image_duplicate_dismissed":0,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[1],"tags":[782308494],"class_list":["post-3719","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","tag-linear-algebra"],"jetpack_publicize_connections":[],"jetpack_likes_enabled":true,"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p6Eiku-XZ","jetpack-related-posts":[],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/anagileway.com\/wp-content\/uploads\/2026\/04\/flag-rank.png?fit=2340%2C700&ssl=1","_links":{"self":[{"href":"https:\/\/anagileway.com\/en\/wp-json\/wp\/v2\/posts\/3719","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/anagileway.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/anagileway.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/anagileway.com\/en\/wp-json\/wp\/v2\/users\/24727826"}],"replies":[{"embeddable":true,"href":"https:\/\/anagileway.com\/en\/wp-json\/wp\/v2\/comments?post=3719"}],"version-history":[{"count":53,"href":"https:\/\/anagileway.com\/en\/wp-json\/wp\/v2\/posts\/3719\/revisions"}],"predecessor-version":[{"id":3796,"href":"https:\/\/anagileway.com\/en\/wp-json\/wp\/v2\/posts\/3719\/revisions\/3796"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/anagileway.com\/en\/wp-json\/wp\/v2\/media\/3724"}],"wp:attachment":[{"href":"https:\/\/anagileway.com\/en\/wp-json\/wp\/v2\/media?parent=3719"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/anagileway.com\/en\/wp-json\/wp\/v2\/categories?post=3719"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/anagileway.com\/en\/wp-json\/wp\/v2\/tags?post=3719"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}