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.
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.
As an application, find for and .
Finally, given , show that is invertible and find .