A square matrix P is called a projection matrix if $P = P^T$ and $P^2 = P$. Let $P \in \mathbb{R}^{n \times n}$ be a projection matrix. Show that (i) $I - P$ is a projection matrix. (ii) $||Px||_2^2 + ||(I - P)x||_2^2 = ||x||_2^2$ for any $x \in \mathbb{R}^n$. (iii) P is positive semidefinite. (iv) If $P \neq 0$, then $\lambda_{max}(P) = 1$.
Added by Gary C.
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Step 1: To show that I - P is a projection matrix, we need to show that (I - P) = (I - P)^(T) and (I - P)^(2) = (I - P). Show more…
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