与J(0,2)⊕J(0,2)交换的矩阵

hbghlyj

与$A=\left(\begin{array}{rrrr}
0 & 1 & \\
0 & 0 & \\
& & 0 & 1 \\
& & 0 & 0
\end{array}\right)$交换的矩阵的一般形式是什么呢

hbghlyj

Does A commuting with a block-diagonal matrix B=B₁⊕B₂ implies it is block-diagonal?

Wikipedia: Commuting matrices preserve each other's eigenspaces. \[B=\left(\begin{array}{rrrr} a & b & c & d \\ 0 & a & 0 & c \\ e & f & g & h \\ 0 & e & 0 & g \end{array}\right)\qquad AB=BA\]

Output: True

Output: $B=\left(\begin{array}{rrrr} b_{00} & b_{01} & b_{02} & b_{03} \\ b_{10} & b_{11} & b_{12} & b_{13} \\ b_{20} & b_{21} & b_{22} & b_{23} \\ b_{30} & b_{31} & b_{32} & b_{33} \end{array}\right)\quad AB= \left(\begin{array}{rrrr} b_{10} & b_{11} & b_{12} & b_{13} \\ 0 & 0 & 0 & 0 \\ b_{30} & b_{31} & b_{32} & b_{33} \\ 0 & 0 & 0 & 0 \end{array}\right) \quad BA= \left(\begin{array}{rrrr} 0 & b_{00} & 0 & b_{02} \\ 0 & b_{10} & 0 & b_{12} \\ 0 & b_{20} & 0 & b_{22} \\ 0 & b_{30} & 0 & b_{32} \end{array}\right)$

Czhang271828

https://kuing.cjhb.site/forum.php?mod=viewthread&tid=8537&extra=page=3 早回答过了吧，通解左上方的 4 阶方阵就是啊

hbghlyj

Math 504: Advanced Linear Algebra lec14.tex (26.87 KB, Downloads: 18)

2023-6-9 17:17 Upload

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Proposition 1.1.8. Let $J$ be a Jordan matrix whose Jordan blocks are
\[
J_{n_{1}}\left(  0\right)  ,\ \ J_{n_{2}}\left(  0\right)  ,\ \ \ldots
,\ \ J_{n_{k}}\left(  0\right)  .
\]
Let $B$ be an $n\times n$-matrix, written as a block matrix
\[
B=\left(
\begin{array}
[c]{cccc}%
B\left(  1,1\right)   & B\left(  1,2\right)   & \cdots & B\left(  1,k\right)
\\
B\left(  2,1\right)   & B\left(  2,2\right)   & \cdots & B\left(  2,k\right)
\\
\vdots & \vdots & \ddots & \vdots\\
B\left(  k,1\right)   & B\left(  k,2\right)   & \cdots & B\left(  k,k\right)
\end{array}
\right)  ,
\]
where each $B\left(  i,j\right)  $ is an $n_{i}\times n_{j}$-matrix. Then, $B\in\operatorname*{Cent}J$ if and only if each of the $k^{2}$ blocks $B\left(  i,j\right)  $ is an upper-triangular Toeplitz matrix in the wide sense.

Here, we say that a matrix is an upper-triangular Toeplitz matrix in the wide sense if it has the form $\left(
\begin{array}
[c]{cc}%
0 & U
\end{array}
\right)  $, where $U$ is an upper-triangular Toeplitz (square) matrix and $0$ is a zero matrix, or has the form $\left(
\begin{array}
[c]{c}%
U\\
0
\end{array}
\right)  $, where $U$ is an upper-triangular Toeplitz (square) matrix and $0$ is a zero matrix.

(The zero matrices are allowed to be empty.)

Czhang271828

hbghlyj 发表于 2023-6-9 17:20
Math 504: Advanced Linear Algebra
PDF
Proposition 1.1.8. Let $J$ be a Jordan matrix whose Jordan blo ...

简而言之, 注意到通解\[
\begin{pmatrix}
\lambda  & 1\\
& \lambda
\end{pmatrix} \cdot \begin{pmatrix}
0 & a & b\\
0 & 0 & a
\end{pmatrix} =\begin{pmatrix}
0 & a & b\\
0 & 0 & a
\end{pmatrix} \cdot \begin{pmatrix}
\lambda  & 1 & \\
& \lambda  & 1\\
&  & \lambda
\end{pmatrix},
\]以及($\lambda\neq \mu$)\[\begin{pmatrix}
\lambda  & 1\\
& \lambda
\end{pmatrix} \cdot \begin{pmatrix}
0 & 0 & 0\\
0 & 0 & 0
\end{pmatrix} =\begin{pmatrix}
0 & 0 & 0\\
0 & 0 & 0
\end{pmatrix} \cdot \begin{pmatrix}
\mu  & 1 & \\
& \mu  & 1\\
&  & \mu
\end{pmatrix}.\]即可. 之后按照分块矩阵写.