403/503- Homework 5

April 22, 2011

This set is due the last day of lecture, Friday May 6.

Let $f:{\mathbb C}\to{\mathbb C}$ be an entire function,

$f(x)=\displaystyle\sum_{k=0}^\infty a_k x^k$,

where the series converges for all complex numbers $x$.

Basic results about power series give us that the series converges absolutely, i.e.,

$\displaystyle\sum_{k=0}^\infty |a_k| |x|^k<\infty$

for all $x$, and that for any $t>0$, if $S=\displaystyle\sum_{k=0}^\infty b_k$ is a series such that $|b_k|<|a_k|t^k$, then $S$ converges as well.

Given a finite dimensional inner product space $V$, and a $T\in{\mathcal L}(V)$, we want to define $f(T)$, in a way that it is again a linear operator on $V$. The most common example is when $f(x)=e^x$. This “exponential matrix” has applications in differential equations and elsewhere.

To make sense of $f(T)$, we define making use of the power series of $f$:

$\displaystyle f(T)=\sum_{k=0}^\infty a_k T^k.$

Of course, the problem is to make sure that this expression makes sense. (Use the results of Homework 4 to) show that this series converges, and moreover

$\displaystyle \|f(T)\|_1\le\sum_{k=0}^\infty |a_k|\|T\|_1^k.$

Fixing a basis for $V$, suppose that $T$ is diagonal. Compute $f(T)$ in that case. In particular, in ${\mathbb C}^3$, find $e^A$ where

$\displaystyle A=\left(\begin{array}{ccc}2&0&0\\ 0&-1&0\\ 0&0&5\end{array}\right).$

Show that, in general, the computation of $f(T)$ reduces to the computation of $f(A)$ for $A$ a matrix in Jordan canonical form.

For

$\displaystyle A=\left(\begin{array}{ccccc} \lambda&1&0&\cdots&0\\ 0&\lambda&1&\cdots&0\\ 0&0&\lambda&\cdots&0\\ &&&\cdots\\ 0&0&0&\cdots&\lambda\end{array}\right)$

a Jordan block, show that in order to actually find $f(A)$ reduces to finding formulas for $A^n$ for $n=0,1,\dots$ Find this formula, and use it to find a formula for $f(A)$. It may be useful to review the basics of Taylor series for this.

As an application, find $e^A$ for $A=\displaystyle \left(\begin{array}{cc}1&1\\ 1&1\end{array}\right)$ and $A=\displaystyle \left(\begin{array}{ccc}7/6&2/3&1/6\\ -5/3&1/3&7/3\\ -9/2&0&9/2\end{array}\right)$.

Finally, given $T\in{\mathcal L}(V)$, show that $e^T$ is invertible and find $\det(e^T)$.