403/503- Homework 5

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 $\det(e^T).$

This entry was posted on Friday, April 22nd, 2011 at 2:43 pm and is filed under 403/503: Linear Algebra II. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

3 Responses to 403/503- Homework 5

andrescaicedo says:

April 22, 2011 at 7:08 pm

I changed the last matrix to have a non-diagonalizable example.

Reply
Rachel Byrd says:

April 30, 2011 at 11:08 am

Hello Dr. Caicedo,

When we’re supposing that T is diagonal, do you mean the matrix associated to T is diagonal? Are we still computing f(T) or then f(M(T))? I think I’m a little confused as to when we’re using the matrices or the linear operators in the function f.

thanks,

Reply
- andrescaicedo says:
  
  May 1, 2011 at 10:07 am
  
  Hi Rachel: Yes; once we fix a basis $B$ , we can identified $T$ with the matrix $M={\mathcal M}(T,B)$ , and by saying that $T$ is diagonal I meant that $M$ is.
  
  Reply

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