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I am confused about why the orthogonal projection of matrices onto subspaces is given by a change-of-basis-like formula. For example, in the below image from these notes, why is the orthogonal projection of matrix A onto the subspace $V_m$ given by $V_m^* A V_m$?

From https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.12.5265&rep=rep1&type=pdf

When I search online, I come across a formula like From https://en.wikipedia.org/wiki/Projection_(linear_algebra)#Projection_matrix (for example, from the Wikipedia article), which does not look at all like $V_m^* A V_m$.

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    $\begingroup$ What's written in the Wikipedia article is the projection matrix. $P_A$ is the matrix that projects a given vector onto the subspace spanned by the columns of $A$. $\endgroup$ Commented Feb 18, 2021 at 13:00

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I have the same question, but don't have the reputation to comment.

It's worth noting that you have two different $A$ matrices in your question - the $A$ in the standard projection formula corresponds to your $V_m$. Because the column-vectors of the subspace are orthonormal, $V_m^T V_m = I$, and so the projection matrix (in this notation) is $P \equiv V_m V_m^T$. Here is where I get stuck.

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