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

Let be an entire function,

,

where the series converges for all complex numbers .

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

for all , and that for any , if is a series such that , then converges as well.

Given a finite dimensional inner product space , and a , we want to define , in a way that it is again a linear operator on . The most common example is when . This “exponential matrix” has applications in differential equations and elsewhere.

To make sense of , we define making use of the power series of :

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

Fixing a basis for , suppose that is diagonal. Compute in that case. In particular, in , find where

Show that, in general, the computation of reduces to the computation of for a matrix in Jordan canonical form.

For

a Jordan block, show that in order to actually find reduces to finding formulas for for Find this formula, and use it to find a formula for . It may be useful to review the basics of Taylor series for this.

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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.

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

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,

Hi Rachel: Yes; once we fix a basis , we can identified with the matrix , and by saying that is diagonal I meant that is.