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Is it true that projection is a normal matrix? It's clear that orthogonal projection is, but what about non-orthogonal projection?

By normal matrix, I mean matrix A such that $AA' = A'A$.

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  • Eigenspaces of a non-orthogonal projection aren't orthogonal and hence non-orthogonal projection isn't a normal operator. – Matcha Latte Jul 06 '21 at 09:25

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Not in general. Here's a counter-example:

Let $A = \begin{bmatrix} 0 & 0 \\ 2 & 1 \\ \end{bmatrix}$. You can easily show $A^2 = A$, so it is a projection matrix.

However, $AA^T = \begin{bmatrix} 0 & 0 \\ 0 & 5 \\ \end{bmatrix} \neq A^TA = \begin{bmatrix} 4 & 2 \\ 2 & 1 \\ \end{bmatrix}$

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If fact, every non-orthogonal projection $A$ is non-normal. Indeed, the adjoint $A'$ is also a projection, and $$\operatorname{ran}A' = (\ker A)^\perp \ne \operatorname{ran}A$$ since $A$ is not orthogonal. If two projections commute, they must have the same range.