91影视

Consider the matrix$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& 鈥�& a_{1n}\\ a_{21}& a_{22}& 鈰�& a_{2n}\\ 鈰�& 鈰�& 鈰�& 鈰�\\ a_{m1}& a_{m2}& 鈰�& a_{mn}\end{array}\right]$ where$a_{ij}$ are real.

Simplify the equation $A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& 鈥�& a_{1n}\\ a_{21}& a_{22}& 鈰�& a_{2n}\\ 鈰�& 鈰�& 鈰�& 鈰�\\ a_{m1}& a_{m2}& 鈰�& a_{mn}\end{array}\right]$as follows.

role="math" localid="1659418385609" $$\begin{gathered} A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& 鈥�& a_{1n}\\ a_{21}& a_{22}& 鈰�& a_{2n}\\ 鈰�& 鈰�& 鈰�& 鈰�\\ a_{m1}& a_{m2}& 鈰�& a_{mn}\end{array}\right] \\ A=\left[\begin{array}{cccc}{a}_{11}& 0& 鈰�& 0\\ 0& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right]+\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 0& a_{22}& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right]+\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ a_{21}& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right]+..+\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 0& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& a_{mn}\end{array}\right] \\ A=a_{11}\left[\begin{array}{cccc}1& 0& 鈰�& 0\\ 0& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right]+a_{22}\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 0& 1& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right]+a_{21}\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 1& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right]+..+a_{mn}\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 0& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 1\end{array}\right] \end{gathered}$$

In the matrix $A,\left\{\left[\begin{array}{cccc}1& 0& 鈰�& 0\\ 0& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right],\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 0& 1& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right],\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 1& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right].,,.\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 0& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 1\end{array}\right]\right\}$are linear independent.

Therefore, the matrix A is spanned by $\mathrm{span}\left\{\left[\begin{array}{cccc}1& 0& 鈰�& 0\\ 0& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right],\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 0& 1& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right],\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 1& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right].,,.\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 0& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 1\end{array}\right]\right\}$.

Hence, the dimension of A is mn and spanned by $\mathrm{span}\left\{\left[\begin{array}{cccc}1& 0& 鈰�& 0\\ 0& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right],\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 0& 1& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right],\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 1& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 0\end{array}\right].,,.\left[\begin{array}{cccc}0& 0& 鈰�& 0\\ 0& 0& 鈰�& 0\\ 鈰�& 鈰�& 鈰�& 鈰�\\ 0& 0& 鈰�& 1\end{array}\right]\right\}$.