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Linear Algebra

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Eigenvalues of Orthogonal Projections

LINALG-1F2HG3

Let $P$ be the matrix of the orthogonal projection of $\mathbb{R}^n$ onto a $m$ dimensional subspace $S\subset \mathbb{R}^n$. Suppose that $\{u_k\}_{k=1}^n$ in an orthonormal basis for $\mathbb{R}^n$ such that $\{u_k\}_{k=1}^m$ is an orthonormal basis for $S$.

Which of these four statements is correct?

A

The eigenvalues of $P$ are $1$ and $0$ with multiplicity $m$ and $n-m$, respectively.

B

The eigenvalues of $P$ are $1$ and $0$ with multiplicity $n-m$ and $m$, respectively.

C

$Pu_k=0$ for $k=1, 2, \dots, m$.

D

$Pu_k=u_k$ for $k=m+1, \dots, n$.