91影视

If a linear space has a basis with n elements then all other bases of V, consists of n elements as well.

Also we say that n is the dimension of V

$\mathit{d}\mathit{i}\mathit{m}\mathbf{\left(}\mathbf{V}\mathbf{\right)}\mathbf{=}\mathit{n}$

Consider, B is a diagonal with$3脳3$matrices.

Assume,$B=\left[\begin{array}{l}3\text{鈥夆赌夆赌�}0\text{鈥夆赌夆赌�}0\\ 0\text{鈥夆赌夆赌�}6\text{鈥夆赌夆赌夆��}0\\ 0\text{鈥夆赌夆赌�}0\text{鈥夆赌夆赌夆��}9\end{array}\right]$is the$3脳3$diagonal matrix of B

$\begin{array}{l}|位I-B|=位\left[\begin{array}{l}1\text{鈥夆赌夆赌�}0\text{鈥夆赌夆赌�}0\\ 0\text{鈥夆赌夆赌�}1\text{鈥夆赌夆赌�}0\\ 0\text{鈥夆赌夆赌�}0\text{鈥夆赌夆赌�}1\end{array}\right]-\left[\begin{array}{l}3\text{鈥夆赌夆赌�}0\text{鈥夆赌夆赌�}0\\ 0\text{鈥夆赌夆赌�}6\text{鈥夆赌夆赌�}0\\ 0\text{鈥夆赌夆赌�}0\text{鈥夆赌夆赌�}9\end{array}\right]\\ |位I-B|=0\\ \left[\begin{array}{l}位\text{鈥夆赌夆赌�}0\text{鈥夆赌夆赌�}0\\ 0\text{鈥夆赌夆赌�}位\text{鈥夆赌夆赌�}0\\ 0\text{鈥夆赌夆赌�}0\text{鈥夆赌夆赌�}位\text{鈥夆赌夆赌�}\end{array}\right]-\left[\begin{array}{l}3\text{鈥夆赌夆赌�}0\text{鈥夆赌夆赌�}0\\ 0\text{鈥夆赌夆赌�}6\text{鈥夆赌夆赌�}0\\ 0\text{鈥夆赌夆赌�}0\text{鈥夆赌夆赌�}9\end{array}\right]=0\\ \left[\begin{array}{l}位-3\text{鈥夆赌夆赌夆�夆赌�}0\text{鈥夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌夆��}0\\ 0\text{鈥夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌�}位-6\text{鈥夆赌夆赌夆�夆赌夆赌�}0\\ 0\text{鈥夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌�}0\text{鈥夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌�}位-9\text{鈥夆赌夆赌�}\end{array}\right]=0\end{array}$

Further simplify as follows

$\begin{array}{l}(位-3\text{鈥�})(位-6\text{鈥�})(位-9\text{鈥�})=0\\ 位=3,6\text{鈥夆赌�}or\text{鈥�}9\end{array}$

Thus, the possible dimensions of the space V of all$3脳3$matrices A that commute with B are$位=0,3,6\text{鈥夆赌�}or\text{鈥�}9$.

Since the vector of the matrix A in the diagonal is$A\stackrel{鈫�}{c}=\stackrel{鈫�}{0}$.

Hence the solution.