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\homework{5: Inner Products and Real Symmetric Matrices}{Due Wednesday, February 19, by 1:30pm}


Homework problems need to show work and contain proofs in order to receive full credit.  Simply stating an answer is only half of the problem in mathematics; you also need to include an argument that persuades your audience that your answer is correct!  If you have any questions, feel free to contact either Yihan or I via email or office hours.  Have fun!

\begin{enumerate}
\item Given a matrix $A$, we say that $B$ is a \textbf{cubic root} of $A$ if $B^3 = A$.  
\begin{enumerate}
\item Suppose that $A$ is a real-valued symmetric $n \times n$ matrix.  Prove that $A$ has a cubic root.
\item Is this true for nonsymmetric matrices?  Either find a counterexample, or prove that this is true for all matrices.
\end{enumerate}
\item Suppose that $A$ is a real-valued $2 \times 2$ matrix.  For any two vectors $\vec{v}, \vec{w} \in \mathbb{R}^2$, define the ``$A$-product'' of $\vec{v}$ and $\vec{w}$ as follows:
\begin{align*}
\langle \vec{v}, \vec{w} \rangle_A = (A\vec{v}) \cdot \vec{w}.
\end{align*}
On HW\#3, you studied this object for a specific value of $A$.  
\begin{enumerate}
\item Show that for $A = \begin{bmatrix} 2 & 1 \\ 1 & 2\end{bmatrix}$, this $A$-product is an inner product, as defined in class.
\item In general, suppose $A$ is a real-valued symmetric matrix.  Is this always an inner product?  If so, prove it; if not, create a counterexample, and come up with conditions on $A$ that will insure that this $A$-product is an inner product.
\end{enumerate}
\item Suppose that $A, B$ are a pair of complex-valued $n \times n$ unitary matrices.  Prove that $AB$ is a unitary matrix.
\item The \textbf{trace} of a $n \times n$ matrix $A$, denoted $\textrm{tr}(A)$, is the sum of the entries on its diagonal.  For example, the trace of $\begin{bmatrix} 1 & 2 \\ 4 & 5\end{bmatrix}$ is $1+5 = 6$.  \\\\
Prove that for any two $n \times n$ matrices $A, B$, we have $\textrm{tr}(AB) = \textrm{tr}(BA).$
\end{enumerate}





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