Do Diagonal Matrices Always Commute?

Let A be an n×n matrix and let Λ be an n×n diagonal matrix. Is it always the case that AΛ=ΛA? If not, when is it the case that AΛ=ΛA?

If we restrict the diagonal entries of Λ to being the equal (i.e. Λ=drag(a,a,,a)), then it is clear that AΛ=AaI=aIA=ΛA. However, I can’t seem to come up with an argument for the general case.

Answer

It is possible that a diagonal matrix Λ commutes with a matrix A when A is symmetric and AΛ is also symmetric. We have

ΛA=(AΛ)=(AΛ)=AΛ

The above trivially holds when A and Λ are both diagonal.

Attribution
Source : Link , Question Author : Jeff Scott , Answer Author : user322903

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