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\begin{document}

\begin{center}
{\bf Math 601 -- Fall 2026 -- Harry Tamvakis}
\end{center}


\begin{center}
{\bf PROBLEM SET 3 -- Due February 26, 2026}
\end{center}



\skipline
\noin
{\bf A1)} Prove that every real symmetric matrix has a real
symmetric cube root; i.e., if $A$ is real symmetric, there is a 
real symmetric $B$ such that $B^3=A$. Characterize those
real symmetric matrices that have a real symmetric
square root.

\skipline
\noin
{\bf A2)} Find all values of the real number $a$ for which the
quadratic form 
\[
\dis
Q(x,y,z)=x^2+y^2+z^2+2axy+2ayz
\]
is positive definite. 


\skipline \noin {\bf A3)} Let $V$ be a finite dimensional vector space
and $B:V\times V \to F$ be a non-degenerate alternating bilinear form
on $V$. In class we proved that $\dim(V)=2n$ is even. A subspace $W$
of $V$ is called {\em isotropic} if $B(x,y)=0$ for all $x,y\in
W$. Prove that if $W$ is isotropic then $0\leq \dim W \leq n$.  Give
an example of an isotropic subspace $W$ with $\dim(W)=n$.

 

\skipline \noin {\bf B1)} Let $T$ be a normal operator on a finite
dimensional complex inner product space $V$. Let $\l_1,\ldots, \l_k$
be the distinct eigenvalues of $T$. Let $W_i$ be the eigenspace of
$\l_i$ and $E_i:V\ra V$ the orthogonal projection of $V$ onto
$W_i$. You assume the spectral theorem for $T$, which was proved in
class.

\medskip
\noin
(a) Prove that we have
\begin{equation}
\label{equ}
T=\l_1E_1+\cdots+\l_kE_k=\sum_{i=1}^k\l_iE_i.
\end{equation}
The decomposition (\ref{equ}) is called the {\em spectral resolution} of $T$. 


\medskip
\noin
(b) The orthogonal projections $E_i$ in (\ref{equ}) are canonically
associated with $T$; in fact, they are polynomials in $T$. Show 
that if we define polynomials $e_i$ by
\[
\dis
e_i(x)=\prod_{j\neq i}\left(\frac{x-\l_j}{\l_i-\l_j}\right),
\]
then $E_i=e_i(T)$ for $1\leq i\leq k$.

\medskip
\noin 
(c) Let $S=\{\l_1,\ldots,\l_k\}$ be the set of eigenvalues of $T$;
$S$ is called the {\em spectrum} of $T$.  If $f:S\ra \C$ is any
function, the linear operator $f(T)$ is defined by the equation
\[
f(T):=\sum_{i=1}^kf(\l_i)E_i.
\]
Prove that $f(T)$ is a normal operator whose spectrum is $f(S)$.

\medskip
\noin 
(d) Suppose that $S:V\to V$ is a linear operator such that $ST=TS$.
Prove that $ST^*=T^*S$.


 
\skipline \noin {\bf B2)} Let $V$ be a finite dimensional vector space
over a field $F$ and suppose that $B:V\times V\to F$ is a bilinear
form on $V$. We say that $B$ is {\em symmetric} if $B(x,y)=B(y,x)$ for
any $x,y\in V$ and $B$ is {\em alternating} if $B(x,x)=0$ for any
$x\in V$. Assume that whenever $x,y\in V$ are such that $B(x,y)=0$,
then $B(y,x)=0$ (that is, the orthogonality relation on $V$ induced by
$B$ is symmetric).  Prove that either $B$ is symmetric or $B$ is
alternating.


 
\skipline \noin {\bf B3)} Let $V$ be a vector space of dimension $2n$
over a field $F$ and choose a basis $e_1,\ldots,e_{2n}$ of $V$. To
each skew-symmetric matrix $A=\{a_{ij}\}\in M_{2n}(F)$ assign the
element $\omega := \sum_{i<j} a_{ij} \,e_i \wedge e_j$ in
$\Lambda^2V$, and then to $\omega$ assign $\wedge^n\omega = f(A)\,
e_1\wedge \cdots \wedge e_{2n}$ in $\Lambda^{2n}V$. The function
$A\mapsto f(A)$ does not depend on the choice of basis of $V$. Suppose
that $e'_1,\ldots,e'_{2n}$ is a different basis with $e'_j=\sum_i
p_{ij}e_i$, and let $P$ be the invertible matrix $\{p_{ij}\}$.

\medskip
\noin (a) Show that $\sum_{i<j}a_{ij} \,e_i\wedge e_j =
\sum_{i<j}b_{ij}\,e'_i\wedge e'_j$ where the matrix $B:=\{b_{ij}\}$
satisfies $A=PBP^t$.  Deduce that $f(PBP^t)=(\det P)f(B)$.

\medskip
\noin (b) If $A$ is invertible, show that $A=PJP^t$ for some matrix
$P$, where $J:=\mathrm{diag}(S,\ldots,S)$ with $\dis
S:=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right)$. Deduce
that $f(A) = (\det P) f(J)$ and $\det A = (\det P)^2 = (f(A)/f(J))^2$.

\medskip
\noin (c) Prove that $\dis f(A) = n!
\sum_I\text{sgn}(I)\,a_{i_1i_2}a_{i_3 i_4}\cdots a_{i_{2n-1}i_{2n}}$
summed over all partitions $I$ of the set $\{1,\ldots,2n\}$ into $n$
pairs $\{i_k,i_{k+1}\}$ where $i_k < i_{k+1}$ and $\text{sgn}(I)$ is
the sign of the permutation $\dis \left(\begin{array}{cccc} 1 & 2&
  \cdots & 2n \\ i_1 & i_2 & \cdots & i_{2n}\end{array}\right)$. In
particular, $f(J)=n!$. Conclude with an explicit formula for $\Pf(A)
:= f(A)/f(J)$.



\skipline
\medskip


\begin{center}
{\bf \underline{Extra Credit Problems}}
\end{center}


\skipline \noin {\bf C1)} Suppose that $A=\{a_{ij}\}_{1\leq i,j\leq n}$
is a symmetric matrix in $M_n(\R)$.  Prove that $A$ is positive
definite if and only if for each $m\in [1,n]$, we have
\[
\det\left(\{a_{ij}\}_{1\leq i,j \leq m}\right) > 0.
\]


\medskip \noin {\bf C2)} Let $A$ be a symmetric positive definite 
$n\times n$ real matrix, $x:=(x_1,\ldots,x_n)$, and $\langle\ \, ,\  \rangle$
denote the usual inner product on $\R^n$.
Evaluate the integral
\[
\int_{\R^n}e^{-\left<Ax,x\right>}\,dx_1\cdots dx_n.
\]




\end{document}
