|
|
Math 504: Advanced Linear Algebra
lec14.tex
(26.87 KB, 下载次数: 0)
PDF
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
发表于 2023-6-9 12:22
