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\centerline{Mathematics 244 -- Linear Algebra, section 1}
\centerline{Problem Set 10 -- {\bf Due}: Wednesday, May 2, no later than 
5:00pm}
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\noindent
I.  For each of the following symmetric matrices of $A$,
\item{(i)} Determine the eigenvalues
\item{(ii)} Find an orthonormal basis of ${\bf R}^n$ consisting of eigenvectors
of the matrix
\item{(iii)}  Give the spectral decomposition of $A$
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\item{A)}  $$A = \pmatrix{3 & 5\cr 5 & 3}$$
\item{B)}  $$A = \pmatrix{16 & -4\cr -4 & 1}$$
\item{C)}  $$A = \pmatrix{3 & 1 & 1\cr 1 & 3 & 1\cr 1 & 1 & 3\cr}$$
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\noindent
II.  Let $A \in M_{n\times n}({\bf R})$, with entries $a_{ij}$,
and let $p(\lambda) = \det(A - \lambda I)$ be its characteristic polynomial.
\item{A)}  Show by induction on $n$ that 
$$p(\lambda) = (a_{11} - \lambda)(a_{22} - \lambda)\cdot \cdots \cdot
(a_{nn} - \lambda) + ({\rm terms\ of\ degree\ } \le n-2) $$
(where degree means degree as a polynomial in the variable $\lambda$).  
\item{B)}  Deduce from part $A$ that the coefficient of $\lambda^{n-1}$
in $p(\lambda)$ is $(-1)^{n-1}(a_{11} + a_{22} + \cdots + a_{nn})$.
This scalar, the sum of the diagonal entries in $A$, is called
the {\it trace} of $A$ and written ${\rm Tr}(A)$.
\item{C)}  Show that if $A$ has $n$ real eigenvalues $\lambda_1,\ldots,\lambda_n$ (not necessarily distinct), then ${\rm Tr}(A) = \lambda_1 + \cdots + \lambda_n$
and $\det(A) = \lambda_1 \cdot \cdots \cdot \lambda_n$.  
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\noindent
III.  An $n\times n$ matrix $Q$ is said to be {\it orthogonal}
if $Q$ is invertible and $Q^{-1} = Q^t$.  
\item{A)}  Show that if $Q$ is orthogonal, then the columns of 
$Q$ form an orthonormal basis of ${\bf R}^n$.
\item{B)}  Show conversely that if the columns of $Q$ form an
orthonormal basis for ${\bf R}^n$, then $Q$ is orthogonal.
\item{C)}  Show that if $P$ and $Q$ are orthogonal, then $PQ$ is
also orthogonal.
\item{D)}  Show that if $Q$ is orthogonal, then $Q^{-1}$ is
also orthogonal.  
\bye
