Linear Algebra – Representing Orthogonal Projections of Linear Operators

Question

I sat on a lecture given recently where the presenter denoted the orthogonal projection of a matrix $A$ (on $\mathbb{R}^n$) onto a subspace as $P_nAP_n$ where $P_n$ is the orthogonal projection onto the subspace of $\mathbb{R}^n$. I understand how to project a vector onto a subspace, but am still unsure how the composition of those matrices gives the orthogonal projection of $A$ onto the subspace.

Answer 1

I think I have figured this out. Given vectors {$v_1 ... v_p$}, we can compute the projection of $A$ onto the subspace spanned by the vectors ($\mathbb{R}^p$) via the matrix

$ A = \begin{bmatrix} <Av_1,v_1> & ... & <Av_1,v_p> \\ \vdots \\ <Av_p,v_1> & ... & <Av_p,v_p> \end{bmatrix}. $

Then such a matrix of inner products can be represented as $P^TAP$.