Matrix multiplication question (diagonal matrices)

471 Views Asked by At

Suppose $AB = BA$ and $A^2+B^2 = I$, where A and B are complex matrices.

My feeling is that this implies that both A and B are diagonal matrices. But I'm having trouble proving it.

Appreciate any help.

2

There are 2 best solutions below

0
On BEST ANSWER

Consider the matrices $A=\binom{0\ 0}{0\ 0}$ and $B=\binom{0\ 1}{1\ 0}$.

2
On

Let $A=0$ and $B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\right)$, this provides a counterexample.