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).

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