Diagonalize matrix multiplication

Question

Let $V$ be the space of $2\times 2$ matrices with complex coefficients. Let $A \in V$ and let $L_A:V \to V$, defined by $L_A(X)=A\cdot X$. I am trying to solve the exercise (10) from this book: find a basis in $V$ such that the $4x4$ matrix of $L_A$ is block diagonal i.e. is of the form $$\left(\begin{array}{cc} A & 0 \\ 0 & B\end{array}\right).$$ With $A$ and $B$ $2\times 2$ matrices. The linear map $L_A$ is diagonalisable as a map from $\mathbb{C}^4 \to \mathbb{C}^4$, but I'm not sure how to obtain the required form and besides, the eigenvalues look rather ugly.

Answer 1

It's difficult to recommend a systematic approach here. You might come across the answer by noting that the set $$ \left\{\pmatrix{x_1&0\\x_2&0}: x_1,x_2 \in \Bbb C\right\} $$ is an invariant subspace. We can say the same for $\{xy^T : x \in \Bbb C^2\}$ for any fixed $y \in \Bbb C^2$.

Answer:

Consider the basis $$\left\{\pmatrix{1&0\\0&0}, \pmatrix{0&0\\1&0}, \pmatrix{0&1\\0&0}, \pmatrix{0&0\\0&1}\right\}.$$ Note that the order of the basis is important.