Pick favorite colors and download full screen images. One is in Japanese.

As a lifetime math lover, I’m very glad to see that Prof. Gilbert Strang recently published his new book, “Linear Algebra for Everyone.”

This book is the most easy-to-read sequel to his legend “Introduction to Linear Algebra.” He reconstructed the contents aligning **the** **5 factorization of a matrix** and **the big picture(4 subspaces) of Linear Algebra** as described in “A 2020 Vision of Linear Algebra“(where **6 factorization** is introduced). For more information for these books and concepts;

- Linear Algebra for Everyone – http://math.mit.edu/everyone/
- Introduction to Linear Algebra – http://math.mit.edu/linearalgebra/
- A 2020 Vision of Linear Algebra – https://ocw.mit.edu/resources/res-18-010-a-2020-vision-of-linear-algebra-spring-2020/

As a big fan of the both of these books, I tried categorization of matrices which appear in his work, with examples, key properties and the factorizations. And the above drawing is what I came up with!

*Legend*

*A General Matrix**C Independent Column Matrix**in A = CR**R Reduced Row Echelon form in A = CR*,*Upper Triangular Gram-Schmidt factors(QR)**S Symmetric Matrix**Q Orthogonal Matrix**L Lower Triangular Matrix**U Upper Triangular Matrix in LU or**Orthonormal Basis Matrix of column space in SVD**P Permutation Matrix in PA = LU or Projection Matrix**Λ Diagonal Matrix with eigenvalues**Σ Diagonal Matrix with singular values**V Orthonormal Basis Matrix of row space in SVD**J Jordan Form Matrix**I Identity Matrix**O Zero Matrix**𝜆 Eigenvalue*

*Six Factorizations of Matrix*

*A = CR (Column Row) Independent Columns in C**PA = LU (LU ) Lower and Upper Triangular**A = QR(Gram-Schmidt) Orthogonal Columns in Q with Upper triangular R**A = XΛX*^{-1}*(Diagonalization)**Eigenvalues in Λ,**Eigenvectors in X**S = Q**Λ*Q^{T}*(Diagonalization) Eigenvalues in Λ*,*Orthogonal Eigenvectors in Q**A=UΣV*^{T}*(SVD) Singular values in Diagonal**Σ*,*Orthonormal vectors in U, V*

Here’s thoughts behind the drawing.

- All
**Matrices**(*m×n*) can be factorized to*A=CR*(independent columns and rows to see the column rank quals the row rank). - All
**Matrices**(*m×n*) can be factorized to*A=UΣV*(SVD) with orthonormal basis^{T}*U*and*V*in each space. **Square**Matrices(*n×n*) are either**Invertible**or**Singular**.- Invertibility can be checked by whether
*A=LU*has full**pivots**,*det(A) ≠ 0*, or all the**eigenvalues**are none-zero. - An
**Invertible**Matrix is factorized to*A=QR*using**Orthogonal**Matrix*Q*(Gram-Schmidt). - A
**Square**Matrix is either**Diagonalizable**or not. If not, still**Jordan-decomposable**. - A
**Diagonalizable**Matrix is factorized to*A=XΛX*^{–}^{1}with a**Diagonal**Matrix*Λ*with**eigenvalues**and an**Invertible**matrix*X*. - A Non-diagonalizable Matrix is factorized to
*A=XJX*^{-1}with a**Jordan**form Matrix*J*and an**Invertible**matrix*X*. - A Matrix is
**Diagonalizable**as*QΛQ*by an^{T}**Orthogonal**Matrix when and only when it is**Normal***(A*^{T}*A=AA*^{T}). **Symmetric**Matrices (*S=S*^{T}) and**Orthogonal**Matrices (*Q*^{–}^{1}*=Q*^{T}) are typical and important**Normal**Matrices. (Another is**Anti-Symmetric**which is not expressed in the drawing)- There are some matrix types both Symmetric and Orthogonal. They include Householder, Hadamard and Identity. (not in the drawing)
- All
**eigenvalues**of an**Orthogonal**Matrix are*|λ|=1*. - All
**eigenvalues**of a**Symmetric**Matrix are real. - A
**Symmetric**Matrix is called “**Positive Semidefinite”**if all eigenvalues*λ≥0*. - A
**Symmetric**Matrix is called “**Positive Definite”**if all eigenvalues*λ>0*. **Projection**Matrices(*P*^{2}*= P = P*^{T}) are**Symmetric**and**Positive Semidefinite**.- All
**Eigenvalues**of a**Projection**Matrix are*λ=1 or 0*. - A
**Projection**Matrix to the subspace spanned by column vectors of*A*is*P=A(A*^{T}*A)*^{-1}*A*^{T}. (not expressed in the drawing) *A*^{T}*A*for any matrix A is**Positive Semidefinite**, and even**Positive Definite**if and only if the columns of A are independent.- If a Matrix is
**Invertible**, its inverse is expressed as*A*^{-1}*=VΣ*^{-1}U^{T}. - Even a Matrix is not
**Invertible**and not even**Square**, there exists a pseudo inverse*A*^{–}^{1}*=VΣ*^{-1}*U*^{T} - To extend to Complex Matrices; (not expressed in the drawing)
**Orthogonal**should be extended to**Unitary****Symmetric**should be extended to**Hermitian***A*^{T }should be extended to*A*^{H}(conjugated and transposed)

- Lastly,
**Jordan**is placed in Singular world in this map, but it is sometimes invertible, and other times singular(pity that I couldn’t find a better place to put it).

To start this project, I emailed to Prof. Strang, first to confirm the concept, then exchanged more than 20 emails, and finally finished it today. Thank you Prof. Gilbert Strang for the persistent support and up-cheers.

– from personal email Set.29, 2020

Thank you Kenji ! I can see that this figure might be interesting to a lot of people. Well done ! This is a great success. I hope you enjoyed this creation.

Very best wishes Gil

God, I really respect him as a great teacher and educator! I first met him on YouTube, MIT 18.06 lecture in OpenCourseWare and got inspired so much that I even made a T-shirt of his Big Picture(4 subspaces) of Linear Algebra.

That led me to the conversations with him and the drawing. For the record, I retained the first version of the drawing by hand.

Any feedback is welcome!

This is really great. Thanks!

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Oh, thanks for the comment! You may like this too. https://github.com/kenjihiranabe/The-Art-of-Linear-Algebra/blob/main/The-Art-of-Linear-Algebra.pdf

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