Do Diagonal Matrices Always Commute?

Question 1

Let $A$ be an $n \times n$ matrix and let $\Lambda$ be an $n \times n$ diagonal matrix. Is it always the case that $A\Lambda = \Lambda A$ ? If not, when is it the case that $A \Lambda = \Lambda A$ ?

If we restrict the diagonal entries of $\Lambda$ to being the equal (i.e. $\Lambda = \text{drag}(a, a, \dots, a)$ ), then it is clear that $A\Lambda = AaI = aIA = \Lambda A$ . However, I can’t seem to come up with an argument for the general case.

Question 2

It is possible that a diagonal matrix $\Lambda$ commutes with a matrix $A$ when $A$ is symmetric and $A \Lambda$ is also symmetric. We have

$\Lambda A = (A^{\top}\Lambda^\top)^{\top} = (A\Lambda)^\top = A\Lambda$

The above trivially holds when $A$ and $\Lambda$ are both diagonal.

Do Diagonal Matrices Always Commute?

Answer

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