Question 4.1.17: Find a basis of the space V of all matrices B that commute w......
Find a basis of the space V of all matrices B that commute with A =\begin{bmatrix}0&1\\2&3\end{bmatrix}. see Example 13.
We need to find all matrices B =\begin{bmatrix}a&b\\c&d\end{bmatrix} such that
\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}0&1\\2&3\end{bmatrix}=\begin{bmatrix}0&1\\2&3\end{bmatrix}\begin{bmatrix}a&b\\c&d\end{bmatrix}.
The entries of B must satisfy the linear equations
2b = c, a + 3b = d, 2d = 2a + 3c, c + 3d = 2b + 3d.
The last two equations are redundant, so that the matrices B in V are of the form
B=\begin{bmatrix}a&b\\2b&a+3b\end{bmatrix}=a\begin{bmatrix}1&0\\0&1\end{bmatrix}+b\begin{bmatrix}0&1\\2&3\end{bmatrix}=aI_2+bA.
Since the matrices I_2 and A are linearly independent, a basis of V is
(I_2,A)=\left(\begin{bmatrix}1&0\\0&1\end{bmatrix},\begin{bmatrix}0&1\\2&3\end{bmatrix}\right).