9.6, where we use sums and I get really excited about sums

Section 9.6 is entitled “The Exponential of a Matrix.”

Allow me to warn you up front that this section is chock-full of propositions and theorems and such nonsense, so be aware quite a bit of this summary will be quotes. I apologize only because summaries should be more of an “in your own words” kind of thing, and that’s not the case sometimes. However, I hesitate to apologize only because the authors are paid money to explain math things the best, and I am not paid to write this blog. Therefore, I think I’m allowed to use direct quotes now and then.

Let’s now discuss what we do when we have an eigenvalue where the geometric multiplicity does not equal the algebraic multiplicity. When n = 1, we have an initial value problem

From there, we looked for exponential solutions to the system x’ = Ax of the form x(t) = e^λtv. This worked as long as λ was an eigenvalue of A and v was the associated eigenvector. Most of the time, this has worked beautifully and we’re happy and can move on with our lives. However, we have moments and examples where this doesn’t work out and we’re not happy anymore. So our goal is to find a term called the exponential of a matrix, which will give us solutions to the equation x’ = Ax.

In order to define the exponential of a matrix, we actually have to look at the series for the exponential function (which I had to look at for my physics homework last week, so I’m really exciting and I’m kind of geeking out about it. Don’t judge me. No but seriously I'm really excited about this). Therefore, the exponential of the matrix A is defined to be

Just for clarification, A² = A*A, A³ = A*A*A, etc. Also, we have I as the first term because A⁰ is I. Since A is defined to be an n × n matrix, e^A will also be an n × n matrix (provided this series converges). Since we’re dealing with a series, we must consider the partial sum matrices

The book says that series in the definition (i.e. the infinite series I’ve shown you) will converge for every matrix A. They don’t prove it, and I don’t want to prove it, so I guess we’ll all have to take their word for it.

Here are some fun facts from the book: “The exponential of a diagonal matrix is the diagonal matrix containing the exponentials of the diagonal entries. In particular, we have

When r = 0, we have the important special case

 Unfortunately, there is no easy formula for the exponential of a matrix that is not diagonal” (417).

Well, that last fact wasn’t fun at all, was it?

So here’s a proposition for you: “Suppose A is an n × n matrix.

1. Then

2. If v ∈ Rⁿ, the function x(t) = e^tAv is a solution to the initial value problem x’ = Ax with x(0) = v” (418).

In other words, we’ll be able to solve the initial value problem if we can compute e^tAv.

Since we’re dealing with a sum, we could potentially have a finite sum very easily if a majority of the terms turn out to be 0. This leads us to the next proposition for an n × n matrix A:

“1. If Av = 0, then e^tAv = v for all t.

2. If A²v = 0, then e^tAv = v + tAv for all t.

3. More generally, if A^kv = 0, then

for all t” (419).

This property is called truncation (as in the infinite sum is truncated to a finite sum). This means we can compute e^tAv when v is in the nullspace of A (or in the nullspace of a power of A).

If we look at the properties of these exponentials, we’ll see we get one of them from a law of exponents, mainly that e^a+b = e^ae^b. However, this isn’t the case for every pair of matrices (let’s call them A and B).

Since we learned in day one of matrices that it’s rarely true for AB = BA (in math words, we would say that A and B commute). However, I do not fear for the future, since this is a textbook and the authors are most likely paid money to find examples that work.

Another property of these exponentials we’ll be looking at is one to do with singularity, namely “if A is an n × n matrix, then e^A is a nonsingular matrix whose inverse is e^-A” (420).

Another proposition for you: “Suppose A is an n × n matrix, λ is a number, and v is an n-vector.

1. If [A – λI]v = 0, then e^tAv = e^λtv for all t.

2. If [A – λI]²v = 0, then e^λt(v + t[A – λI]v) for all t.

3. More generally, if k is a positive integer and [A – λI]^kv = 0, then

for all t” (421).

Recall that [A – λI]v = 0 if and only if λ is an eigenvalue and v is the associated eigenvector. The solution in this case would be y(t) = e^tAv = e^λtv which agrees what we learned previously in the chapter. Hooray consistency!

A definition: “If λ is an eigenvector of A, and [A – λI]^pv = 0 for some integer p ≥ 1, we will call v a  generalized eigenvector” (422).

And now for a theorem: “Suppose λ is an eigenvector of A with algebraic multiplicity q. Then there is an integer p ≤ q such that the dimension of the nullspace of [A – λI]^p is equal to q” (422).

So you may be wondering why λ is now an eigenvector. Well, I have no idea, because I originally thought it was a typo. Well, they either made the same typo in two different places, or they meant to make λ an eigenvector. I don’t know.

In any event, we now have a procedure for solving linear systems.

“1. Find the smallest integer p such that the nullspace of [A – λI]^p has dimension q.

2. Find a basis for {v₁, v₂,…, v_q} of the nullspace of [A – λI]^p.

3. For each v_j, 1 ≤ j ≤ q, we have the solution…” (423).

Well, that’s all the learning for section 9.6! I’m not online right now so once I get online I’ll definitely look for websites with examples. I feel like general infinite sums only go so far before you need to know how to deal with them when you enter the real world of dealing with examples. There are probably tons of examples out there, so I’m not worried.

EDIT: I found a couple of websites, some not really related to this section, which might help you out with eigenvalues and eigenvectors and such.

More of an overview: http://tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx and http://www.sosmath.com/diffeq/system/linear/eigenvalue/eigenvalue.html

Real eigenvalues: http://tutorial.math.lamar.edu/Classes/DE/RealEigenvalues.aspx and http://www.sosmath.com/diffeq/system/linear/eigenvalue/real/real.html

Complex eigenvalues: http://www.sosmath.com/diffeq/system/linear/eigenvalue/complex/complex.html

Matrix exponential stuff: http://www.sosmath.com/matrix/expo/expo.html and http://faculty.atu.edu/mfinan/4243/section611.pdf and http://www.math.wisc.edu/~dummit/docs/diffeq_3_systems_of_linear_diffeq.pdf (starts at the very bottom of the eighth page)

In any event, I’ll see you in 9.7.