Chapter 7.7, Problem 8E

Exercise $6$ shows that eigenvectors of a symmetric matrix belonging to distinct eigenvalues are orthogonal. We now show that if $A$ is a symmetric $\left(n\times n\right)$, then $A$ is has a set of $n$ orthogonal eigenvectors. Let $\left\{u_1,u_2,\mathrm{...},u_k\right\}$ be a set of $k$ eigenvectors for $A$, $1\le k\le n$, where $u_i{}^T{u}_j=0$, $i\ne j$. Let $W$ be the subset of $R^n$ defined by

$W=\left\{x:x^T{u}_i=0,i=1,2,\mathrm{...},k\right\}$

From the Gram-Schmidt theorem, the subset $W$ contains nonzero vectors.

a) Prove that $W$ is a subspace of $R^n$.

b) Suppose that $A$ is $\left(n\times n\right)$ and symmetric. Prove that $Ax$ is in $W$ whenever $x$ is in $W$. From Exercise $7$, $A$ has an eigenvector, $u$, in $W$. If we label $u$ as $u_{k+1}$, then by construction $\left\{u_1,u_2,\mathrm{...},u_k,u_{k+1}\right\}$ is a set of $n$ orthogonal eigenvectors, $u_1$, $u_2$,…, $u_n$. Using these, we can form $Q$ so that $Q^{T}AQ=D$, where $D$ is diagonal and $Q^{T}Q=I$.