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\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Weekly Homework \#3}
\date{Assigned Wednesday, 4 October 2017; due Wednesday, 11 October 2017}

\maketitle

\begin{problab}{W3.1}
Let $V,W$ be finite-dimensional inner product spaces.  
\begin{enumalph}
\item Let $\phi\colon V \to V$ be a self-adjoint linear operator.  Recall that $\psi\colon V \to V$ is \emph{positive semidefinite} if $\langle \psi(x),x\rangle \geq 0$ whenever $x \neq 0$.  Show that $\phi$ is positive semidefinite if and only if all of the eigenvalues of $\phi$ are nonnegative.  
\item Now let $\phi\colon V \to W$ be linear.  Show that $\phi^*\phi$ and $\phi\phi^*$ are positive semidefinite.  
\item Show that $\rk(\phi^*\phi)=\rk(\phi\phi^*)=\rk(\phi)$.
\end{enumalph}
\end{problab}

\begin{problab}{W3.2}
Let $V=\bfR^n$ be the standard inner product space.  Let
\[ S=\{x \in V: \| x \|^2=1 \} \]
be the $(n-1)$-dimensional sphere in $V$.  

\begin{enumalph}
\item Suppose that $x,y \in S$ have $\langle x,y \rangle=0$.  Show that $\cos(t)x+\sin(t)y$ lies on $S$ for all $t \in \bfR$.
\item Let $\phi :V \to V$ be a self-adjoint linear map.  By vector calculus, the function $x \mapsto \langle x,\phi (x)\rangle$ achieves a maximum at some point $p \in S$: briefly explain why.  Let $y \in S$ satisfy $\langle p,y \rangle=0$.  Consider the function
\[ f(t) = \langle \cos(t)p+\sin(t)y,\phi (\cos(t)p+\sin(t)y) \rangle. \]
Show that $\langle p,\phi (y)\rangle=0$.  
\item Let $W=\opspan(\{p\})$.  Show that $W^\perp$ is $\phi $-invariant and then conclude that $W$ is $\phi $-invariant.  Conclude that $p$ is an eigenvector!  
\item Parlay the argument of \textup{(c)} into an inductive proof that $V$ has an orthonormal basis of vectors that are eigenvectors for $\phi $.  
\end{enumalph}
\emph{[Note: This argument inductively gives a different ``physical'' or ``geometric'' proof that $\phi$ has an orthonormal basis of eigenvectors: we find an eigenvector by maximizing $\phi$ on the sphere!]}
\end{problab}

\begin{problab}{W3.3}
In each part, let $\phi \colon V \to V$ be the projection on the subspace $W_1$ along the subspace $W_2$, where $V=W_1 \oplus W_2$.  

\begin{enumalph}
\item Show that $\phi$ is an orthogonal projection (i.e., $W_2=W_1^\perp$) if and only if $\| \phi(x) \| \leq \| x\|$ for all $x \in V$.   \emph{[Hint: Let $w_1 \in W_1$ and $w_2 \in W_2$ be nonzero and $c \in \R$, and let $w=cw_1+w_2$.  Show that
\[ 2c \repart\langle w_1,w_2 \rangle + \| w_2 \|^2 \geq 0. \]
If $\langle w_1,w_2 \rangle \neq 0$, derive a contradiction by a choice of $c$; conclude that $\langle w_1,w_2 \rangle =0$ for all $w_1,w_2$.]}

\item What can you conclude if $\phi$ is unitary?  [So don't confuse a projection that is orthogonal with an orthogonal projection!]

\item Suppose that $\phi$ is \emph{normal} over $F=\bfC$.  Prove that $\phi$ is an orthogonal projection.
\end{enumalph}
\end{problab}

\begin{problab}{W3.4}
Let $\phi,\psi \colon V \to V$ be normal operators on a finite-dimensional complex inner product space $V$.  Suppose that $\phi\psi=\psi\phi$.  Prove that there exists an orthonormal basis for $V$ consisting of (simultaneous) eigenvectors for $\phi$ and $\psi$.
\end{problab}

\end{document}