Diagonal matrices are not necessarily square?

Question

I have seen it implied, in the context of singular value decomposition, that diagonal matrices are not necessarily square. Is this true? How can it be true? Can someone please explain this in more detail?

Answer 1

The singular value decomposition of an $m \times n$ matrix $A$ is $A = U \Sigma V^\top$ where $U$ is $m \times m$ orthogonal, $V$ is $n \times n$ orthogonal, and $\Sigma$ is $m \times n$ diagonal ($\Sigma_{ij} = 0$ if $i \ne j$).

This is the only context where I have encountered non-square diagonal matrices. Otherwise, typically when one says "diagonal matrix" one usually assumes it is square, as copper.hat mentioned in his comment.