Thanks for this fantastic resource! One small issue in the MatrixWorld diagram:

MatrixWorld-v1.4.2 defines permutation matrices P as:

• Orthogonal matrices for which all $\lambda$ are roots of 1

This is necessary but not sufficient for P to be a permutation matrix. Two examples of matrices satisfying this property that are not permutation matrices:

• $Q = \begin{bmatrix} -1 \end{bmatrix}$ has eigenvalue $-1$
• $Q = \begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} - 1 & 0 \end{bmatrix}$, the example provided of an orthogonal matrix that is not a permutation matrix, has eigenvalues $\pm i$

This can be fixed by defining permutation matrices P as the intersection of:

• Orthogonal matrices for which all $\lambda$ are roots of 1, with
• Nonnegative matrices, i.e. matrices for which all elements are $\ge 0$

The equivalence follows from https://en.wikipedia.org/wiki/Nonnegative_matrix#Inversion: "The inverse of a non-negative matrix is usually not non-negative. The exception is the non-negative monomial matrices", where a monomial matrix is defined to have "the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column." Hence an orthogonal nonnegative matrix must be a non-negative monomial matrix, and an orthogonal nonnegative matrix with all eigenvalues roots of unity must be a permutation matrix.

A typo in Illustratitions-j.pptx and toc are fixed. In my environment (macOS psutils and ps2eps installed with MacPorts), I need further changes in the options for psselect and ps2eps.

translating a simplified Chinese version, the translated parts are:

• README (see README-zh-CN.md)
• the text content of 5 pages in the main part (see The-Art-of-Linear-Algebra-zh-CN.pdf)

I'm still in the process of translating!

translating a simplified Chinese version, the latest translated parts:

• MapofEigenvalues-zh-CN.png
• MatrixWorld-zh-CN.png
• 5-Factorizations-zh-CN.png

And they appeare in:

• The-Art-of-Linear-Algebra-zh-CN.pdf
• README-zh-CN.md
• MapofEigenvalues-v1.1.pdf
• MapofEigenvalues-v1.1.pptx
• MatrixWorld.pdf
• MatrixWorld.pptx

all translated parts:

• README (see README-zh-CN.md)
• the text content and 3 main patterns in the main part (see The-Art-of-Linear-Algebra-zh-CN.pdf)

translating a simplified Chinese version, the latest translated parts:

• the text content form page 6 to the end in the main part (see The-Art-of-Linear-Algebra-zh-CN.pdf)

all translated parts:

• README (see README-zh-CN.md)
• the text content in the main part (see The-Art-of-Linear-Algebra-zh-CN.pdf)

Thank you, Kenji, for the awesome resource! This PR is a small fix to properly show MathJax formulas. README.Rmd is the document knitted to README.md (output: github_document); the latter will contain rendered formulas. Example of the result

Hello Mr. Hiranabe, I found Matrix World from Prof. Strang's website. I just note two typos:

1. should be "diagonalizable" in "diagonizable by orthogonal matrix" in the "Normal" section.
2. should be "permutation in "permutaion of I" in the "Permutation" section.

Thank you for the beautiful visualizations. Calvin

on macOS, I installed MacTex, but psselect still not availble. it looks like it's not compatible on macOS.

what platform and toolchain did you use to generate PDFs?

Appreciated for your efforts.

Dear Mr. Hiranabe,

I found two (small) typo's in your github document

1. Sec. 3.5 (p.124) Dimentions (-> dimensions) of the four subspaces
2. This (-> is) the spectral theorem (below Fig. 16).

Wonderfull visualizations!

If I may suggest one idea: you number the columns, rows in some images. Why not also number the dots?

Yes, I realize: the way Gilbert Strang makes us look at matrices, columns, rows, operations is the first (for me: giant) step in understanding linear algebra. In my view, your images make a very compact summarization of it. Thank you!

Jan Baltussen, The Netherlands

Great pdf, thanks.

I think the sentence "The four subspaces consists of N(A) + C(A^T) (which are perpendicular to each other) in R^n and N(A^T)

• N(A) in R^m" on page 3 should be "... N(A^T) + C(A) ..." , shouldn't it?