kenjihiranabe / the-art-of-linear-algebra Goto Github PK

修正第3页上的英文大小写

修正前：

\begin{itemize}
\item 1.1节 (p.3) LInear combinations
\item 1.3节 (p.21) Matrices and Column Spaces
\end{itemize}

修正后：

\begin{itemize}
\item 1.1节 (p.3) Linear combinations
\item 1.3节 (p.21) Matrices and Column Spaces
\end{itemize}

修正第8页上图13无法显示中文

修正前：

\caption{$A的递归秩1矩阵分离$}

修正后：

\caption{$A$的递归秩1矩阵分离}

修正第9页上的中文名词

修正前：

一个对乘矩阵$S$通过一个正交矩阵$Q$和它的转置矩阵, 对角化为$\Lambda$.

修正后：

一个对称矩阵$S$通过一个正交矩阵$Q$和它的转置矩阵, 对角化为$\Lambda$.

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.

I could not find license information - can you add a license? Thanks.

Prof. Strang,

Hello, again, I have one question.

Attached, here's Problem 6 in Pset7.2 in the "Introduction 6" and
"everyone"(the same section number, and the problem number), and its
answer in the solution manual.

I highlighted by a red circle. Why F3(x, y)=x^2+y^2 is positive semidefinite
(in the sense of linear algebra definition) ?
There was a mention in the "everyone" solution, but just "So" is there.
and I'm not convinced. There was no mention in the "introduction" solution.

And also the problem asks for the singular values and vectors, but no
mention in "introduction". "everyone" mentions a rough estimation and
a suggestion to the computational solution.

Thank you, Professor Strang, I'm always so happy to learn new things in LA.
So beautiful. (LA is linear algebra, not Los Angeles)

Best,
-Kenji

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

How do you think about the change?

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?

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.

此处应为秩1投影矩阵

In, black and white version of 3.1, rhs and lhs colums are not distinguishable by colors.

In section 6.4, a symmetric matrix S (对称矩阵S) seems to be typed as 对乘矩阵S. This appears on line 578 of The-Art-of-Linear-Algebra-zh-CN.tex

此处“行””列”写反了，如下图

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