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## Main Question or Discussion Point

I have two matrices which commute, one of which is definitely diagonal:

[itex]\textbf{B}diag\{\underline{\lambda}\} = diag\{\underline{\lambda}\}\textbf{B}[/itex]

and I want to know what I can say about [itex]\textbf{B}[/itex] and/or [itex]\underline{\lambda}[/itex]. Specifically, I feel that either one or both of the following must be correct:

(1) [itex]diag\{\underline{\lambda}\}[/itex] is proportional to identity.

(2) [itex]\textbf{B}[/itex] is diagonal.

[ignoring the trivial cases where one or both matrices equal the zero matrix]

But are there other cases when these two matrices can commute? i.e. Is it possible for both [itex]\textbf{B}[/itex] to be non-diagonal and the elements of [itex]\underline{\lambda}[/itex] to not all be identical?

[itex]\textbf{B}diag\{\underline{\lambda}\} = diag\{\underline{\lambda}\}\textbf{B}[/itex]

and I want to know what I can say about [itex]\textbf{B}[/itex] and/or [itex]\underline{\lambda}[/itex]. Specifically, I feel that either one or both of the following must be correct:

(1) [itex]diag\{\underline{\lambda}\}[/itex] is proportional to identity.

(2) [itex]\textbf{B}[/itex] is diagonal.

[ignoring the trivial cases where one or both matrices equal the zero matrix]

But are there other cases when these two matrices can commute? i.e. Is it possible for both [itex]\textbf{B}[/itex] to be non-diagonal and the elements of [itex]\underline{\lambda}[/itex] to not all be identical?