GILBERT STRANG: OK, thanks. Here's a second video
that involves the matrix exponential. But it has a new idea
in it, a basic new idea. And that idea is two matrices
being called "similar." So that word "similar"
has a specific meaning, that a matrix A, is
similar to another matrix B, if B comes from A this way. Notice this way. It means there's some matrix M--
could be any invertible matrix. So that I take A,
multiply on the right by M and on the
left by M inverse. That'd probably give
me a new matrix. Call it B. That matrix
is called "similar" to B. I'll show you examples of
matrices that are similar. But first is to get
this definition in mind. So in general, a
lot of matrices are similar to-- if I have a certain
matrix A, I can take any M, and I'll get a similar
matrix B. So there are lots of similar matrices. And the point is all
those similar matrices have the same eigenvalues. So there's a little
family of matrices there, all similar
to each other and all with the same eigenvalues. Why do they have the
same eigenvalues? I'll just show you, one line. Suppose B has an
eigenvalue of lambda. So B is M inverse AM. So I have this. M inverse AMx is lambda x. That's Bx. B has an eigenvalue of lambda. I want to show that A has
an eigenvalue of lambda. OK. So I look at this. I multiply both sides
by M. That cancels this. So when I multiply by M,
this is gone, and I have AMx. But the M shows up on
the right-hand side, I have lambda Mx. Now I would just look
at that, and I say, yes. A has an eigenvector-- Mx
with eigenvalue lambda. A times that vector is
lambda times that vector. So lambda is an eigenvalue of A.
It has a different eigenvector, of course. If matrices have
the same eigenvalues and the same eigenvectors,
that's the same matrix. But if I do this,
allow an M matrix to get in there, that
changes the eigenvectors. Here they were
originally x for B. And now for A,
they're M times x. It does not change the
eigenvalues because of this M on both sides allowed
me to bring M over to the right-hand side
and make that work. OK. Here are some similar matrices. Let me take some. So these will be all similar. Say 2, 3, 0, 4. OK? That's a matrix A. I can see
its eigenvalues are 2 and 4. Well, I know that it will be
similar to the diagonal matrix. So there is some matrix
M that connects this one with this one, connects
this A with that B. Well, that B is really capital lambda. And we know what matrix
connects the original A to its eigenvalue matrix. What is the M that does that? It's the eigenvector matrix. So to get this particular--
to get this guy, starting from here, I use
M is V for this example to produce that. Then B is lambda. But there are other
possibilities. So let me see. I think probably a
matrix is-- there is the matrix, A transpose. Is that similar to A? Is A transpose similar to A? Well, answer-- yes. The transpose matrix has those
same eigenvalues, 2 and 4, and different eigenvectors. And those eigenvectors
would connect the original A and this A or that A transpose. So the transpose of a matrix
is similar to the matrix. What about if I
change the order? 4, 0, 0, 2. So I've just flipped
the 2 and the 4, but of course I haven't
changed the eigenvalues. You could find the
M that does that. You can find an M so that
if I multiply on the right by M and on the left by M
inverse, it flips those. So there's another
matrix similar. Oh, there could be plenty more. All I want to do is have
the eigenvalues be 4 and 2. Shall I just create some more? Here is a 0, 6. I wanted to get the trace right. 4 plus 2 matches 0 plus 6. Now I have to get the
determinant right. That has a determinant of 8. What about a 2 and
a minus 4 there? I think I've got the
trace correct-- 6. And I've got the
determinant correct-- 8. And there the determinant is 8. So that would be
a similar matrix. All similar matrices. A family of similar matrices
with the eigenvalues 4 and 2. So I want to do another
example of similar matrices. What will be different
in this example is there'll be
missing eigenvectors. So let me say, 2, 2, 0, 1. So that has eigenvalues 2 and
2 but only one eigenvector. Here is another
matrix like that. Say, so the trace should be 4. The determinant should be 4. So maybe I put a 2
and a minus 2 there. I think that has the
correct trace, 4, and the great
determinant, also 4. So that will have eigenvalues 2
and 2 and only one eigenvector, so it's similar to this. Now here's the point. You might say, what
about 2, 2, 0, 0. That has the
correct eigenvalues, but it's not similar. There's no matrix M that
connects that diagonal matrix with these other matrices. That matrix has no
missing eigenvectors. These matrices have one
missing eigenvector. What's called the Jordan form. The Jordan form. So that didn't belong. That's not in that family. The Jordan form is--
you could say-- well, that'll be the Jordan form. The most beautiful member of
the family is the Jordan form. So I have a whole lot of
matrices that are similar. That is the most beautiful,
but it's not in the family. It's related but
not in the family. It's not similar to those. And the best one
would be this one. So the Jordan form would be
that one with the eigenvalues on the diagonal. But because there's a
missing eigenvector, there has to be a
reason for that. And it's in the 1 there,
and I can't have a 0 there. OK. So that's the idea
of similar matrices. And now I do have one more
important note, a caution about matrix exponentials. Can I just tell you this
caution, this caution? If I look at e to the
A times e to the B. The exponential of A times
the exponential of B. My caution is that usually that
is not e to the B, e to the A. If I put B and A in
the opposite order, I get something different. And it's also not
e to the A plus B. Those are all different. Which, if I had 1 by 1, just
numbers here, of course, that's the great rule
for exponentials. But for matrix exponentials,
that rule doesn't work. That is not the same
as e to the A plus B. And I can show you why. e to the A is I plus A plus
1/2 A squared and so on. e to the B is I plus B plus
1/2 B squared and so on. And I do that multiplication. And I get I. And I get an
A. And I get a B times I. And now I get 1/2 B squared
and an AB and 1/2 A squared. Can I put those down? 1/2 A squared, and
there's an A times a B. And there's a 1/2 B squared. OK. This makes the point. If I multiply the
exponentials in this order, I get A times B. What
if I multiply them in the other order,
in that order? If I multiply e to the
B times e to the A, then the B's will be
out in front of the A's. And this would become a
BA, which can be different. So already I see that
the two are different. Here is e to the
A, e to the B. It has A before B. If
I do it this way, it'll have B before A.
If I do it this way, it'll have a mixture. So e to the A plus B
will have a I and an A and a B and a 1/2
A plus B squared. So that'll be 1/2 of A squared
plus AB plus BA plus B squared. Different again. Now I have a sort of
symmetric mixture of A and B. In this case, I had A
before B. In this case, I had B on the left side of A. So all three of
those are different, even in this term
of the series that defines those exponentials. And that means that
systems of equations, if the coefficients change over
time, are definitely harder. We were able to solve
dy dt equals, say, cosine of t times y. Do you remember how-- that
this was solvable for a 1 by 1. We put the exponent--
the solution was y is e to the-- we
integrated cosine t and got sine t times y of 0. e to the sine t-- Can I just think of putting
that into the differential equation-- its derivative. The derivative of e to the
sine t will be e to the sine t. I'm using the chain rule. The derivative of e to the
sine t will be e to the sine t again, times the derivative
of sine t, which is cos t, so it works. That's fine as a solution. But if I have matrices
here-- if I have matrices, then the whole thing goes wrong. You could say that the
chain rule goes wrong. You can't put the
integral up there and then take the derivative
and expect it to come back down. The chain rule will not work
for matrix exponentials, the simple chain rule. And the fact is
that we don't have nice formulas for the
solutions to linear systems with time-varying coefficients. That has become a harder problem
when we went from one equation to a system of an equation. So this is the caution slide
about matrix exponentials. They're beautiful. They work perfectly if you
just have one matrix A. But if somehow two
matrices are in there or a bunch of
different matrices, then you lose the good rules,
and you lose the solution. OK. Thank you.
