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We are going to need a few
facts about fundamental
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matrices, and I am worried that
over the weekend this spring
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activities weekend you might
have forgotten them.
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So I will just spend two or
three minutes reviewing the most
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essential things that we are
going to need later in the
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period.
What we are talking about is,
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I will try to color code things
so you will know what they are.
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First of all,
the basic problem is to solve a
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system of equations.
And I am going to make that a
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two-by-two system,
although practically everything
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I say today will also work for
end-by-end systems.
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Your book tries to do it
end-by-end, as usual,
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but I think it is easier to
learn two-by-two first and
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generalize rather than to wade
through the complications of
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end-by-end systems.
So the problem is to solve it.
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And the method I used last time
was to describe something called
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a fundamental matrix.
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A fundamental matrix for the
system or for A,
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whichever you want,
remember what that was.
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That was a two-by-two matrix of
functions of t and whose columns
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were two independent solutions,
x1, x2.
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These were two independent
solutions.
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In other words,
neither was a constant multiple
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of the other.
Now, I spent a fair amount of
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time showing you the two
essential properties that a
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fundamental matrix had.
We are going to need those
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today, so let me remind you the
basic properties of X and the
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properties by which you could
recognize one if you were given
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one.
First of all,
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the easy one,
its determinant shall not be
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zero, is not zero for any t,
for any value of the variable.
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That simply expresses the fact
that its two columns are
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independent, linearly
independent, not a multiple of
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each other.
The other one was more bizarre,
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so I tried to call a little
more attention to it.
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It was that the matrix
satisfies a differential
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equation of its own,
which looks almost the same,
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except it's a matrix
differential equation.
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It is not our column vectors
which are solutions but matrices
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as a whole which are solutions.
In other words,
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if you take that matrix and
differentiate every entry,
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what you get is the same as A
multiplied by that matrix you
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started with.
This, remember,
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expressed the fact,
it was just really formal when
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you analyzed what it was,
but it expressed the fact that
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it says that the columns solved
the system.
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The first thing says the
columns are independent and the
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second says each column
separately is a solution to the
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system.
That is as far,
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more or less.
Then I went in another
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direction and we talked about
variation of parameters.
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I am not going to come back to
variation of parameters today.
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We are going in a different
tack.
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And the tack we are going on is
I want to first talk a little
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more about the fundamental
matrix and then,
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as I said, we will talk about
an entirely different method of
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solving the system,
one which makes no mention of
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eigenvalues or eigenvectors,
if you can believe that.
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But, first, the one confusing
thing about the fundamental
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matrix is that it is not unique.
I have carefully tried to avoid
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talking about the fundamental
matrix because there is no "the"
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fundamental matrix,
there is only "a" fundamental
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matrix.
Why is that?
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Well, because these two columns
can be any two independent
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solutions.
And there are an infinity of
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ways of picking independent
solutions.
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That means there is an infinity
of possible fundamental
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matrices.
Well, that is disgusting,
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but can we repair it a little
bit?
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I mean maybe they are all
derivable from each other in
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some simple way.
And that is,
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of course, what is true.
Now, as a prelude to doing
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that, I would like to show you
what I wanted to show you on
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Friday but, again,
I ran out of time,
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how to write the general
solution --
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-- to the system.
The system I am talking about
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is that pink system.
Well, of course,
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the standard naďve way of doing
it is it's x equals,
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the general solution is an
arbitrary constant times that
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first solution you found,
plus c2, times another
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arbitrary constant,
times the second solution you
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found.
Okay.
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Now, how would you abbreviate
that using the fundamental
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matrix?
Well, I did something very
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similar to this on Friday,
except these were called Vs.
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It was part of the variation
parameters method,
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but I promised not to use those
words today so I just said
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nothing.
Okay.
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What is the answer?
It is x equals,
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how do I write this using the
fundamental matrix,
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x1, x2?
Simple.
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It is capital X times the
column vector whose entries are
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c1 and c2.
In other words,
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it is x1, x2 times the column
vector c1, c2,
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isn't it?
Yeah.
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Because if you multiply this
think top row,
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top row, top row c1,
plus top row times c2,
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that exactly gives you the top
row here.
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And the same way the bottom row
here, times this vector,
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gives you the bottom row of
that.
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It is just another way of
writing that,
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but it looks very efficient.
Sometimes efficiency isn't a
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good thing, you have to watch
out for it, but here it is good.
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So, this is the general
solution written out using a
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fundamental matrix.
And you cannot use less symbols
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than that.
There is just no way.
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But that gives us our answer
to, what do all fundamental
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matrices look like?
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Well, they are two columns are
solutions.
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The answer is they look like --
Now, the first column is an
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arbitrary solution.
How do I write an arbitrary
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solution?
There is the general solution.
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I make it a particular one by
giving a particular value to
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that column vector of arbitrary
constants like two,
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three or minus one,
pi or something like that.
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The first guy is a solution,
and I have just shown you I can
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write such a solution like X,
c1 with a column vector,
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a particular column vector of
numbers.
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This is a solution because the
green thing says it is.
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And side by side,
we will write another one.
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And now all I have to do is,
of course, there is supposed to
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be a dependent.
We will worry about that in
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just a moment.
All I have to do is make this
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look better.
Now, I told you last time,
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by the laws of matrix
multiplication,
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if the first column is X c1 and
the second column is X c2,
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using matrix multiplication
that is the same as writing it
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this way.
This square matrix times the
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matrix whose entries are the
first column vector and the
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second column vector.
Now, I am going to call this C.
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It is a square matrix of
constants.
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It is a two-by-two matrix of
constants.
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And so, the final way of
writing it is just what
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corresponds to that,
X times C.
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And so X is a given fundamental
matrix, this one,
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that one, so the most general
fundamental matrix is then the
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one you started with,
and multiply it by an arbitrary
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square matrix of constants,
except you want to be sure that
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the determinant is not zero.
Well, the determinant of this
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guy won't be zero,
so all you have to do is make
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sure that the determinant of C
isn't zero either.
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In other words,
the fundamental matrix is not
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unique, but once you found one
all the other ones are found by
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multiplying it on the right by
an arbitrary square matrix of
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constants, which is nonsingular,
it has determinant nonzero in
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other words.
Well, that was all Friday.
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That's Friday leaking over into
Monday.
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And now we begin the true
Monday.
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Here is the problem.
Once again we have our
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two-by-two system,
or end-by-end if you want to be
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super general.
There is a system.
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What do we have so far by way
of solving it?
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Well, if your kid brother or
sister when you go home said,
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a precocious kid,
okay, tell me how to solve this
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thing, I think the only thing
you will be able to say is well,
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you do this,
you take the matrix and then
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you calculate something called
eigenvalues and eigenvectors.
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Do you know what those are?
I didn't think you did,
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blah, blah, blah,
show how smart I am.
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And you then explain what the
eigenvalues and eigenvectors
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are.
And then you show how out of
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those make up special solutions.
And then you take a combination
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of that.
In other words,
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it is algorithm.
It is something you do,
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a process, a method.
And when it is all done,
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you have the general solution.
Now, that is fine for
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calculating particular problems
with a definite model with
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definite numbers in it where you
want a definite answer.
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And, of course,
a lot of your work in
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engineering and science classes
is that kind of work.
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But the further you get on,
well, when you start reading
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books, for example,
or god forbid start reading
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papers in which people are
telling you, you know,
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they are doing engineering or
they are doing science,
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they don't want a method,
what they want is a formula.
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In other words,
the problem is to fill in the
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blank in the following.
You are writing a paper,
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and you just set up some
elaborate model and A is a
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matrix derived from that model
in some way, represents bacteria
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doing something or bank accounts
doing something,
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I don't know.
And you say,
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as is well-known,
the solution is,
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of course, you only have
letters here,
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no numbers.
This is a general paper.
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The solution is given by the
formula.
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The only trouble is,
we don't have a formula.
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All we have is a method.
Now, people don't like that.
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What I am going to produce for
you this period is a formula,
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and that formula does not
require the calculation of any
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eigenvalues, eigenvectors,
doesn't require any of that.
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It is, therefore,
a very popular way to fill in
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to finish that sentence.
Now the question is where is
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that formula going to come from?
Well, we are,
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for the moment,
clueless.
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If you are clueless the place
to look always is do I know
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anything about this sort of
thing?
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I mean is there some special
case of this problem I can solve
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or that I have solved in the
past?
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And the answer to that is yes.
You haven't solved it for a
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two-by-two matrix but you have
solved it for a one-by-one
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matrix.
A one-by-one matrix also goes
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by the name of a constant.
It is just a thing.
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It's a number.
Just putting brackets around it
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doesn't conceal the fact that it
is just a number.
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Let's look at what the solution
is for a one-by-one matrix,
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a one-by-one case.
If we are looking for a general
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solution for the end-by-end
case, it must work for the
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one-by-one case also.
That is a good reason for us
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starting.
That looks like x,
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doesn't it?
A one-by-one case.
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Well, in that case,
I am trying to solve the
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system.
The system consists of a single
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equation.
That is the way the system
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looks.
How do you solve that?
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Well, you were born knowing how
to solve that.
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Anyway, you certainly didn't
learn it in this course.
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You separate variables,
blah, blah, blah,
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and the solution is x equals,
the basic solution is e to the
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at,
and you multiply that by an
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arbitrary constant.
Now, that is a formula for the
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solution.
And it uses the parameter in
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the equation.
I didn't have to know a special
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number.
I didn't have to put a
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particular number here to use
that.
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Well, the answer is that the
same idea, whatever the answer I
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give here has got to work in
this case, too.
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But let's take a quick look as
to why this works.
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Of course, you separate
variables and use calculus.
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I am going to give you a
slightly different argument that
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has the advantage of
generalizing to the end-by-end
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case.
And the argument goes as
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follows for that.
It uses the definition of the
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exponential function not as the
inverse to the logarithm,
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which is where the fancy
calculus books get it from,
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nor as the naďve high school
method, e squared means
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you multiply e by itself and e
cubed means you do it three
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times and so on.
And e to the one-half means you
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do it a half a time or
something.
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So, the naďve definition of the
exponential function.
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Instead, I will use the
definition of the exponential
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function that comes from an
infinite series.
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Leaving out the arbitrary
constant that we don't have to
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bother with.
e to the a t is the series one
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plus at plus a squared t squared
over two factorial.
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I will put out one more term
and let's call it quits there.
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If I take this then argument
goes let's just differentiate
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it.
In other words,
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what is the derivative of e to
the at with respect to t?
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Well, just differentiating term
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by term it is zero plus the
first term is a,
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the next term is a squared
times t.
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This differentiates to t
squared over two factorial.
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And the answer is that this is
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equal to a times,
if you factor out the a,
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what is left is one plus a t
plus a squared t squared over
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two factorial
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In other words,
it is simply e to the at.
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In other words,
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by differentiating the series,
using the series definition of
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the exponential and by
differentiating it term by term,
269
00:17:44 --> 00:17:50
I can immediately see that is
satisfies this differential
270
00:17:48 --> 00:17:54
equation.
What about the arbitrary
271
00:17:50 --> 00:17:56
constant?
Well, if you would like,
272
00:17:52 --> 00:17:58
you can include it here,
but it is easier to observe
273
00:17:56 --> 00:18:02
that by linearity if e to the a
t solves the equation so does
274
00:18:00 --> 00:18:06
the constant times it because
the equation is linear.
275
00:18:05 --> 00:18:11
Now, that is the idea that I am
going to use to solve the system
276
00:18:12 --> 00:18:18
in general.
What are we doing to say?
277
00:18:16 --> 00:18:22
Well, what could we say?
The solution to,
278
00:18:21 --> 00:18:27
well, let's get two solutions
at once by writing a fundamental
279
00:18:28 --> 00:18:34
matrix.
"A" fundamental matrix,
280
00:18:33 --> 00:18:39
I don't claim it is "the" one,
for the system x prime equals A
281
00:18:41 --> 00:18:47
x.
That is what we are trying to
282
00:18:46 --> 00:18:52
solve.
And we are going to get two
283
00:18:51 --> 00:18:57
solutions by getting a
fundamental matrix for it.
284
00:18:57 --> 00:19:03
The answer is e to the a t.
285
00:19:04 --> 00:19:10
Isn't that what it should be?
I had a little a.
286
00:19:07 --> 00:19:13
Now we have a matrix.
Okay, just put the matrix up
287
00:19:11 --> 00:19:17
there.
Now, what on earth?
288
00:19:13 --> 00:19:19
The first person who must have
thought of this,
289
00:19:16 --> 00:19:22
it happened about 100 years
ago, what meaning should be
290
00:19:20 --> 00:19:26
given to e to a matrix power?
Well, clearly the two na•ve
291
00:19:25 --> 00:19:31
definitions won't work.
The only possible meaning you
292
00:19:30 --> 00:19:36
could try for is using the
infinite series,
293
00:19:33 --> 00:19:39
but that does work.
So this is a definition I am
294
00:19:37 --> 00:19:43
giving you, the exponential
matrix.
295
00:19:40 --> 00:19:46
Now, notice the A is a
two-by-two matrix multiplying it
296
00:19:44 --> 00:19:50
by t.
What I have up here is that
297
00:19:46 --> 00:19:52
it's basically a two-by-two
matrix.
298
00:19:49 --> 00:19:55
Its entries involve t,
but it's a two-by-two matrix.
299
00:19:53 --> 00:19:59
Okay.
We are trying to get the analog
300
00:19:56 --> 00:20:02
of that formula over there.
Well, leave the first term out
301
00:20:02 --> 00:20:08
just for a moment.
The next term is going to
302
00:20:05 --> 00:20:11
surely be A times t.
This is a two-by-two matrix,
303
00:20:09 --> 00:20:15
right?
What should the next term be?
304
00:20:12 --> 00:20:18
Well, A squared times t squared
over two factorial.
305
00:20:16 --> 00:20:22
What kind of a guy is that?
Well, if A is a two-by-two
306
00:20:20 --> 00:20:26
matrix so is A squared.
How about this?
307
00:20:23 --> 00:20:29
This is just a scalar which
multiplies every entry of A
308
00:20:28 --> 00:20:34
squared.
And, therefore,
309
00:20:30 --> 00:20:36
this is still a two-by-two
matrix.
310
00:20:33 --> 00:20:39
That is a two-by-two matrix.
This is a two-by-two matrix.
311
00:20:36 --> 00:20:42
No matter how many times you
multiply A by itself it stays a
312
00:20:40 --> 00:20:46
two-by-two matrix.
It gets more and more
313
00:20:43 --> 00:20:49
complicated looking but it is
always a two-by-two matrix.
314
00:20:46 --> 00:20:52
And now I am multiplying every
entry of that by the scalar t
315
00:20:50 --> 00:20:56
cubed over three factorial.
316
00:20:53 --> 00:20:59
I am continuing on in that way.
What I get, therefore,
317
00:20:56 --> 00:21:02
is a sum of two-by-two
matrices.
318
00:21:00 --> 00:21:06
Well, you can add two-by-two
matrices to each other.
319
00:21:03 --> 00:21:09
We've never made an infinite
series of them,
320
00:21:06 --> 00:21:12
we haven't done it,
but others have.
321
00:21:09 --> 00:21:15
And this is what they wrote.
The only question is,
322
00:21:12 --> 00:21:18
what should we put in the
beginning?
323
00:21:15 --> 00:21:21
Over there I have the number
one.
324
00:21:17 --> 00:21:23
But I, of course,
cannot add the number one to a
325
00:21:20 --> 00:21:26
two-by-two matrices.
What goes here must be a
326
00:21:23 --> 00:21:29
two-by-two matrix,
which is the closest thing to
327
00:21:27 --> 00:21:33
one I can think of.
What should it be?
328
00:21:31 --> 00:21:37
The I.
Two-by-two I.
329
00:21:32 --> 00:21:38
Two-by-two identity matrix
looks like the natural candidate
330
00:21:37 --> 00:21:43
for what to put there.
And, in fact,
331
00:21:40 --> 00:21:46
it is the right thing to put
there.
332
00:21:42 --> 00:21:48
Okay.
Now I have a conjecture,
333
00:21:45 --> 00:21:51
you know, purely formally,
changing only with a keystroke
334
00:21:49 --> 00:21:55
of the computer,
all the little a's have been
335
00:21:53 --> 00:21:59
changed to capital A's.
And now all I have to do is
336
00:21:57 --> 00:22:03
wonder if this is going to work.
Well, what is the basic thing I
337
00:22:03 --> 00:22:09
have to check to see that it is
the fundamental matrix?
338
00:22:07 --> 00:22:13
The question is,
I wrote it down all right,
339
00:22:11 --> 00:22:17
but is this a fundamental
matrix for the system?
340
00:22:14 --> 00:22:20
Well, I have a way of
recognizing a fundamental matrix
341
00:22:19 --> 00:22:25
when I see one.
The critical thing is that it
342
00:22:22 --> 00:22:28
should satisfy this matrix
differential equation.
343
00:22:26 --> 00:22:32
That is what I should verify.
Does this guy that I have
344
00:22:32 --> 00:22:38
written down satisfy this
equation?
345
00:22:35 --> 00:22:41
And the answer is,
number two is,
346
00:22:37 --> 00:22:43
it satisfies x prime equals Ax.
347
00:22:41 --> 00:22:47
In other words,
plugging in x equals this e to
348
00:22:45 --> 00:22:51
the at,
whose definition I just gave
349
00:22:49 --> 00:22:55
you.
If I substitute that in,
350
00:22:51 --> 00:22:57
does it satisfy that matrix
differential equation?
351
00:22:56 --> 00:23:02
The answer is yes.
I am not going to calculate it
352
00:23:00 --> 00:23:06
out because the calculation is
actually identical to what I did
353
00:23:04 --> 00:23:10
there.
The only difference is when I
354
00:23:06 --> 00:23:12
differentiated it term by term,
how do you differentiate
355
00:23:10 --> 00:23:16
something like this?
Well, you differentiate every
356
00:23:13 --> 00:23:19
term in it.
But, if you work it out,
357
00:23:15 --> 00:23:21
this is a constant matrix,
every term of which is
358
00:23:18 --> 00:23:24
multiplied by t squared over two
factorial.
359
00:23:21 --> 00:23:27
Well, if you differentiate
every entry of that constant,
360
00:23:25 --> 00:23:31
of that matrix,
what you are going to get is A
361
00:23:27 --> 00:23:33
squared times just the
derivative of that part,
362
00:23:30 --> 00:23:36
which is simply t.
In other words,
363
00:23:34 --> 00:23:40
the formal calculation looks
absolutely identical to that.
364
00:23:40 --> 00:23:46
So the answer to this is yes,
by the same calculation as
365
00:23:45 --> 00:23:51
before, as for the one-by-one
case.
366
00:23:48 --> 00:23:54
And now the only other thing to
check is that the determinant is
367
00:23:55 --> 00:24:01
not zero.
In fact, the determinant is not
368
00:23:59 --> 00:24:05
zero at one point.
That is all you have to check.
369
00:24:04 --> 00:24:10
What is x of zero?
What is the value of the
370
00:24:08 --> 00:24:14
determinant of x is e
to the At?
371
00:24:11 --> 00:24:17
What is the value of this thing
at zero?
372
00:24:14 --> 00:24:20
Here is my function.
If I plug in t equals zero,
373
00:24:18 --> 00:24:24
what is it equal to?
I.
374
00:24:20 --> 00:24:26
What is the determinant of I?
One.
375
00:24:22 --> 00:24:28
It is certainly not zero.
376
00:24:26 --> 00:24:32
377
00:24:38 --> 00:24:44
By writing down this infinite
series, I have my two solutions.
378
00:24:41 --> 00:24:47
Its columns give me two
solutions to the original
379
00:24:44 --> 00:24:50
system.
There were no eigenvalues,
380
00:24:47 --> 00:24:53
no eigenvectors.
I have a formula for the
381
00:24:49 --> 00:24:55
answer.
What is the formula?
382
00:24:51 --> 00:24:57
It is e to the At.
And, of course,
383
00:24:54 --> 00:25:00
anybody reading the paper is
supposed to know what e to the
384
00:24:57 --> 00:25:03
At is.
It means that.
385
00:25:00 --> 00:25:06
This is just marvelous.
There must be a fly in the
386
00:25:03 --> 00:25:09
ointment somewhere.
Only a teeny little fly.
387
00:25:07 --> 00:25:13
There is a teeny little fly
because it is almost impossible
388
00:25:11 --> 00:25:17
to calculate that series for all
reasonable times.
389
00:25:15 --> 00:25:21
However, once in a while it is.
Let me give you an example
390
00:25:20 --> 00:25:26
where it is possible to
calculate the series and were
391
00:25:24 --> 00:25:30
you get a nice answer.
Let's work out an example.
392
00:25:29 --> 00:25:35
393
00:25:36 --> 00:25:42
By the way, you know,
nowadays, we are not back 50
394
00:25:40 --> 00:25:46
years, the exponential matrix
has the same status on,
395
00:25:45 --> 00:25:51
say, a Matlab or Maple or
Mathematica, as the ordinary
396
00:25:51 --> 00:25:57
exponential function does.
It is just a command you type
397
00:25:56 --> 00:26:02
in.
You type in your matrix.
398
00:26:00 --> 00:26:06
And you now say EXP of that
matrix and out comes the answer
399
00:26:04 --> 00:26:10
to as many decimal places as you
want.
400
00:26:06 --> 00:26:12
It will be square matrix with
entries carefully written out.
401
00:26:10 --> 00:26:16
So, in that sense,
the fact that we cannot
402
00:26:13 --> 00:26:19
calculate it shouldn't bother
us.
403
00:26:15 --> 00:26:21
There are machines to do the
calculations.
404
00:26:18 --> 00:26:24
What we are interested in is it
as a theoretical tool.
405
00:26:22 --> 00:26:28
But, in order to get any
feeling for this at all,
406
00:26:25 --> 00:26:31
we certainly have to do a few
calculations.
407
00:26:30 --> 00:26:36
Let's do an easy one.
Let's consider the system x
408
00:26:34 --> 00:26:40
prime equals y,
y prime equals x.
409
00:26:39 --> 00:26:45
This is very easily done by
elimination, but that is
410
00:26:43 --> 00:26:49
forbidden.
First of all,
411
00:26:45 --> 00:26:51
we write it as a matrix.
It's the system x prime equals
412
00:26:50 --> 00:26:56
zero, one, one,
zero, x.
413
00:26:53 --> 00:26:59
Here is my A.
414
00:26:55 --> 00:27:01
And so e to the At
is going to be --
415
00:27:01 --> 00:27:07
A is zero, one,
one, zero.
416
00:27:02 --> 00:27:08
What we want to
417
00:27:05 --> 00:27:11
calculate is we are going to get
both solutions at once by
418
00:27:08 --> 00:27:14
calculating it one fell swoop e
to the At.
419
00:27:12 --> 00:27:18
Okay.
E to the At equals.
420
00:27:13 --> 00:27:19
I am going to actually write
out these guys.
421
00:27:16 --> 00:27:22
Well, obviously the hard part,
the part which is normally
422
00:27:20 --> 00:27:26
going to prevent us from
calculating this series
423
00:27:23 --> 00:27:29
explicitly, by hand anyway,
because, as I said,
424
00:27:26 --> 00:27:32
the computer can always do it.
The value, how do we raise a
425
00:27:32 --> 00:27:38
matrix to a high power?
You just keep multiplying and
426
00:27:37 --> 00:27:43
multiplying and multiplying.
That looks like a rather
427
00:27:41 --> 00:27:47
forbidding and unpromising
activity.
428
00:27:44 --> 00:27:50
Well, here it is easy.
Let's see what happens.
429
00:27:48 --> 00:27:54
If that is A,
what is A squared?
430
00:27:51 --> 00:27:57
I am going to have to calculate
that as part of the series.
431
00:27:56 --> 00:28:02
That is going to be zero,
one, one, zero times zero,
432
00:28:00 --> 00:28:06
one, one, zero,
which is one,
433
00:28:03 --> 00:28:09
zero, zero, one.
434
00:28:06 --> 00:28:12
435
00:28:10 --> 00:28:16
We got saved.
It is the identity.
436
00:28:13 --> 00:28:19
Now, from this point on we
don't have to do anymore
437
00:28:17 --> 00:28:23
calculations,
but I will do them anyway.
438
00:28:21 --> 00:28:27
What is A cubed?
Don't start from scratch again.
439
00:28:26 --> 00:28:32
No, no, no.
A cubed is A squared times A.
440
00:28:30 --> 00:28:36
And A squared is,
441
00:28:34 --> 00:28:40
in real life,
the identity.
442
00:28:35 --> 00:28:41
Of course, you would do all
this in your head,
443
00:28:38 --> 00:28:44
but I am being a good boy and
writing it all out.
444
00:28:41 --> 00:28:47
This is I, the identity,
times A, which is A.
445
00:28:44 --> 00:28:50
I will do one more.
What is A to the fourth?
446
00:28:47 --> 00:28:53
Now, you would be tempted to
say A to the fourth is A
447
00:28:50 --> 00:28:56
squared, which is I times I,
which is I, but that would be
448
00:28:54 --> 00:29:00
wrong.
A to the fourth is A cubed
449
00:28:58 --> 00:29:04
times A,
which is, I have just
450
00:29:02 --> 00:29:08
calculated is A times A,
right?
451
00:29:05 --> 00:29:11
And now that is A squared,
which is the identity.
452
00:29:10 --> 00:29:16
It is clear,
by this argument,
453
00:29:12 --> 00:29:18
it is going to continue in the
same way each time you add an A
454
00:29:18 --> 00:29:24
on the right-hand side,
you are going to keep
455
00:29:22 --> 00:29:28
alternating between the
identity, A, the next one will
456
00:29:27 --> 00:29:33
be identity, the next will be A.
The end result is that the
457
00:29:34 --> 00:29:40
first term of the series is
simply the identity;
458
00:29:39 --> 00:29:45
the next term of the series is
A, but it is multiplied by t.
459
00:29:44 --> 00:29:50
I will keep the t on the
outside.
460
00:29:47 --> 00:29:53
Remember, when you multiply a
matrix by a scalar,
461
00:29:52 --> 00:29:58
that means multiply every entry
by that scalar.
462
00:29:56 --> 00:30:02
This is the matrix zero,
t, t, zero.
463
00:30:00 --> 00:30:06
I will do a couple more terms.
464
00:30:05 --> 00:30:11
The next term would be,
well, A squared we just
465
00:30:08 --> 00:30:14
calculated as the identity.
That looks like this.
466
00:30:12 --> 00:30:18
Except now I multiply every
term by t squared over two
467
00:30:16 --> 00:30:22
factorial.
All right.
468
00:30:19 --> 00:30:25
I'll go for broke.
The next one will be this times
469
00:30:22 --> 00:30:28
t cubed over three factorial.
470
00:30:25 --> 00:30:31
And, fortunately,
I have run out of room.
471
00:30:30 --> 00:30:36
Okay, let's calculate then.
472
00:30:34 --> 00:30:40
473
00:30:54 --> 00:31:00
What is the final answer for e
to At?
474
00:30:57 --> 00:31:03
I have an infinite series of
two-by-two matrices.
475
00:31:00 --> 00:31:06
Let's look at the term in the
upper left-hand corner.
476
00:31:03 --> 00:31:09
It is one plus zero times t
plus one times t squared over
477
00:31:07 --> 00:31:13
two factorial plus zero
times t.
478
00:31:11 --> 00:31:17
It is going to be,
479
00:31:12 --> 00:31:18
in other words,
one plus t squared over two
480
00:31:15 --> 00:31:21
factorial plus
the next term,
481
00:31:18 --> 00:31:24
which is not on the board but I
think you can see,
482
00:31:21 --> 00:31:27
is this.
And it continues on in the same
483
00:31:24 --> 00:31:30
way.
How about the lower left term?
484
00:31:28 --> 00:31:34
Well, that is zero plus t plus
zero plus t cubed over three
485
00:31:32 --> 00:31:38
factorial and so on.
486
00:31:35 --> 00:31:41
It is t plus t cubed over three
factorial plus t to the fifth
487
00:31:39 --> 00:31:45
over five factorial.
488
00:31:42 --> 00:31:48
And the other terms in the
489
00:31:44 --> 00:31:50
other two corners are just the
same as these.
490
00:31:47 --> 00:31:53
This one, for example,
is zero plus t plus zero plus t
491
00:31:51 --> 00:31:57
cubed over three factorial.
492
00:31:55 --> 00:32:01
And the lower one is one plus
zero plus t squared
493
00:31:59 --> 00:32:05
and so on.
This is the same as one plus t
494
00:32:04 --> 00:32:10
squared over two factorial
and so on,
495
00:32:08 --> 00:32:14
and up here we have t plus t
cubed over three factorial
496
00:32:14 --> 00:32:20
and so on.
Well, that matrix doesn't look
497
00:32:18 --> 00:32:24
very square, but it is.
It is infinitely long
498
00:32:22 --> 00:32:28
physically, but it has one term
here, one term here,
499
00:32:27 --> 00:32:33
one term here and one term
there.
500
00:32:31 --> 00:32:37
Now, all we have to do is make
those terms look a little
501
00:32:35 --> 00:32:41
better.
For here I have to rely on the
502
00:32:38 --> 00:32:44
culture, which you may or may
not posses.
503
00:32:42 --> 00:32:48
You would know what these
series were if only they
504
00:32:46 --> 00:32:52
alternated their signs.
If this were a negative,
505
00:32:50 --> 00:32:56
negative, negative then the top
would be cosine t and
506
00:32:55 --> 00:33:01
this would be sine t,
but they don't.
507
00:33:01 --> 00:33:07
So they are the next best
thing.
508
00:33:04 --> 00:33:10
They are what?
Hyperbolic.
509
00:33:06 --> 00:33:12
The topic is not cosine t,
but cosh t.
510
00:33:11 --> 00:33:17
The bottle is sinh t.
511
00:33:15 --> 00:33:21
And how do we know this?
Because you remember.
512
00:33:19 --> 00:33:25
And what if I don't remember?
Well, you know now.
513
00:33:24 --> 00:33:30
That is why you come to class.
514
00:33:29 --> 00:33:35
515
00:33:35 --> 00:33:41
Well, for those of you who
don't, remember,
516
00:33:38 --> 00:33:44
this is e to the t plus e to
the negative t.
517
00:33:44 --> 00:33:50
It should be over two,
but I don't have room to put in
518
00:33:48 --> 00:33:54
the two.
This doesn't mean I will omit
519
00:33:52 --> 00:33:58
it.
It just means I will put it in
520
00:33:55 --> 00:34:01
at the end by multiplying every
entry of this matrix by
521
00:34:00 --> 00:34:06
one-half.
If you have forgotten what cosh
522
00:34:04 --> 00:34:10
t is, it's e to the t plus e to
the negative t divided by two.
523
00:34:09 --> 00:34:15
524
00:34:12 --> 00:34:18
And the similar thing for sinh
t.
525
00:34:14 --> 00:34:20
There is your first explicit
exponential matrix calculated
526
00:34:19 --> 00:34:25
according to the definition.
And what we have found is the
527
00:34:24 --> 00:34:30
solution to the system x prime
equals y,
528
00:34:28 --> 00:34:34
y prime equals x.
A fundamental matrix.
529
00:34:33 --> 00:34:39
In other words,
cosh t and sinh t satisfy both
530
00:34:36 --> 00:34:42
solutions to that system.
Now, there is one thing people
531
00:34:40 --> 00:34:46
love the exponential matrix in
particular for,
532
00:34:44 --> 00:34:50
and that is the ease with which
it solves the initial value
533
00:34:48 --> 00:34:54
problem.
It is exactly what happens when
534
00:34:51 --> 00:34:57
studying the single system,
the single equation x prime
535
00:34:55 --> 00:35:01
equals Ax,
but let's do it in general.
536
00:35:00 --> 00:35:06
Let's do it in general.
What is the initial value
537
00:35:03 --> 00:35:09
problem?
Well, the initial value problem
538
00:35:07 --> 00:35:13
is we start with our old system,
but now I want to plug in
539
00:35:11 --> 00:35:17
initial conditions.
I want the particular solution
540
00:35:15 --> 00:35:21
which satisfies the initial
condition.
541
00:35:18 --> 00:35:24
Let's make it zero to avoid
complications,
542
00:35:22 --> 00:35:28
to avoid a lot of notation.
This is to be some starting
543
00:35:26 --> 00:35:32
value.
This is a certain constant
544
00:35:29 --> 00:35:35
vector.
It is to be the value of the
545
00:35:33 --> 00:35:39
solution at zero.
And the problem is find what x
546
00:35:37 --> 00:35:43
of t is.
Well, if you are using the
547
00:35:41 --> 00:35:47
exponential matrix it is a joke.
It is a joke.
548
00:35:45 --> 00:35:51
Shall I derive it or just do
it?
549
00:35:48 --> 00:35:54
All right.
The general solution,
550
00:35:51 --> 00:35:57
let's derive it,
and then I will put up the
551
00:35:55 --> 00:36:01
final formula in a box so that
you will know it is important.
552
00:36:02 --> 00:36:08
What is the general solution?
Well, I did that for you at the
553
00:36:06 --> 00:36:12
beginning of the period.
Once you have a fundamental
554
00:36:09 --> 00:36:15
matrix, you get the general
solution by multiplying it on
555
00:36:13 --> 00:36:19
the right by an arbitrary
constant vector.
556
00:36:16 --> 00:36:22
The general solution is going
to be x equals e to the At.
557
00:36:20 --> 00:36:26
That is my super fundamental
558
00:36:22 --> 00:36:28
matrix, found without
eigenvalues and eigenvectors.
559
00:36:27 --> 00:36:33
And this should be multiplied
by some unknown constant vector
560
00:36:32 --> 00:36:38
c.
The only question is,
561
00:36:35 --> 00:36:41
what should the constant vector
c be?
562
00:36:38 --> 00:36:44
To find c, I will plug in zero.
When t is zero,
563
00:36:42 --> 00:36:48
here I get x of zero,
here I get e to the A times
564
00:36:47 --> 00:36:53
zero times c.
Now what is this?
565
00:36:51 --> 00:36:57
This is the vector of initial
conditions?
566
00:36:55 --> 00:37:01
What is e to the A times zero?
Plug in t equals zero.
567
00:37:00 --> 00:37:06
What do you get?
I.
568
00:37:04 --> 00:37:10
Therefore, c is what?
c is x zero.
569
00:37:11 --> 00:37:17
It is a total joke.
And the solution is,
570
00:37:17 --> 00:37:23
the initial value problem is x
equals e to the At
571
00:37:26 --> 00:37:32
times x zero.
It is just what it would have
572
00:37:32 --> 00:37:38
been at one variable.
The only difference is that
573
00:37:36 --> 00:37:42
here we are allowed to put the c
out front.
574
00:37:39 --> 00:37:45
In other words,
if I asked you to put in the
575
00:37:41 --> 00:37:47
initial condition,
you would probably write x
576
00:37:44 --> 00:37:50
equals little x zero times e to
the At.
577
00:37:48 --> 00:37:54
And you would be tempted to do
the same thing here,
578
00:37:52 --> 00:37:58
vector x equals vector x zero
times e to the At.
579
00:37:55 --> 00:38:01
Now, you cannot do that.
And, if you try to Matlab will
580
00:38:00 --> 00:38:06
hiccup and say illegal
operation.
581
00:38:02 --> 00:38:08
What is the illegal operation?
Well, x is a column vector.
582
00:38:07 --> 00:38:13
From the system it is a column
vector.
583
00:38:10 --> 00:38:16
That means the initial
conditions are also a column
584
00:38:14 --> 00:38:20
vector.
You cannot multiply a column
585
00:38:17 --> 00:38:23
vector out front and a square
matrix afterwards.
586
00:38:20 --> 00:38:26
You cannot.
If you want to multiply a
587
00:38:23 --> 00:38:29
matrix by a column vector,
it has to come afterwards so
588
00:38:27 --> 00:38:33
you can do zing,
zing.
589
00:38:31 --> 00:38:37
There is no zing,
you see.
590
00:38:33 --> 00:38:39
You cannot put it in front.
It doesn't work.
591
00:38:36 --> 00:38:42
So it must go behind.
That is the only place you
592
00:38:40 --> 00:38:46
might get tripped up.
And, as I say,
593
00:38:43 --> 00:38:49
if you try to type that in
using Matlab,
594
00:38:47 --> 00:38:53
you will immediately get error
messages that it is illegal,
595
00:38:52 --> 00:38:58
you cannot do that.
Anyway, we have our solution.
596
00:38:56 --> 00:39:02
There is our system.
Our initial value problem
597
00:39:00 --> 00:39:06
anyway is in pink,
and its solution using the
598
00:39:04 --> 00:39:10
exponential matrix is in green.
Now, the only problem is we
599
00:39:08 --> 00:39:14
still have to talk a little bit
more about calculating this.
600
00:39:13 --> 00:39:19
Now, the principle warning with
an exponential matrix is that
601
00:39:17 --> 00:39:23
once you have gotten by the
simplest things involving the
602
00:39:21 --> 00:39:27
fact that it solves systems,
it gives you the fundamental
603
00:39:26 --> 00:39:32
matrix for a system,
then you start flexing your
604
00:39:29 --> 00:39:35
muscles and say,
oh, well, let's see what else
605
00:39:33 --> 00:39:39
we can do with this.
For example,
606
00:39:36 --> 00:39:42
the reason exponentials came
into being in the first place
607
00:39:40 --> 00:39:46
was because of the exponential
law, right?
608
00:39:43 --> 00:39:49
I will kill anybody who sends
me emails saying,
609
00:39:46 --> 00:39:52
what is the exponential law?
The exponential law would say
610
00:39:50 --> 00:39:56
that e to the A plus B is equal
to e to the A times e to the B.
611
00:39:54 --> 00:40:00
The law of exponents,
612
00:39:58 --> 00:40:04
in other words.
It is the thing that makes the
613
00:40:01 --> 00:40:07
exponential function different
from all other functions that it
614
00:40:05 --> 00:40:11
satisfies something like that.
Now, first of all,
615
00:40:08 --> 00:40:14
does this make sense?
That is are the symbols
616
00:40:11 --> 00:40:17
compatible?
Let's see.
617
00:40:13 --> 00:40:19
This is a two-by-two matrix,
this is a two-by-two matrix,
618
00:40:16 --> 00:40:22
so it does make sense to
multiply them,
619
00:40:19 --> 00:40:25
and the answer will be a
two-by-two matrix.
620
00:40:21 --> 00:40:27
How about here?
This is a two-by-two matrix,
621
00:40:24 --> 00:40:30
add this to it.
It is still a two-by-two
622
00:40:27 --> 00:40:33
matrix.
e to a two-by-two matrix still
623
00:40:29 --> 00:40:35
comes out to be a two-by-two
matrix.
624
00:40:33 --> 00:40:39
Both sides are legitimate
two-by-two matrices.
625
00:40:37 --> 00:40:43
The only question is,
are they equal?
626
00:40:41 --> 00:40:47
And the answer is not in a
pig's eye.
627
00:40:45 --> 00:40:51
How could this be?
Well, I didn't make up these
628
00:40:50 --> 00:40:56
laws.
I just obey them.
629
00:40:52 --> 00:40:58
I wish I had time to do a
little calculation to show that
630
00:40:58 --> 00:41:04
it is not true.
It is true in certain special
631
00:41:03 --> 00:41:09
cases.
It is true in the special case,
632
00:41:06 --> 00:41:12
and this is pretty much if and
only if, the only case in which
633
00:41:12 --> 00:41:18
it is true is if A and B are not
arbitrary square matrices but
634
00:41:17 --> 00:41:23
commute with each other.
You see, if you start writing
635
00:41:22 --> 00:41:28
out the series to try to check
whether that law is true,
636
00:41:27 --> 00:41:33
you will get a bunch of terms
here, a bunch of terms here.
637
00:41:34 --> 00:41:40
And you will find that those
terms are pair-wise equal only
638
00:41:38 --> 00:41:44
if you are allowed to let the
matrices commute with each
639
00:41:41 --> 00:41:47
other.
In other words,
640
00:41:43 --> 00:41:49
if you can turn AB plus BA into
twice AB then
641
00:41:47 --> 00:41:53
everything will work fine.
But if you cannot do that it
642
00:41:51 --> 00:41:57
will not.
Now, when do two square
643
00:41:53 --> 00:41:59
matrices commute with each
other?
644
00:41:56 --> 00:42:02
The answer is almost never.
It is just a lucky accident if
645
00:42:02 --> 00:42:08
they do, but there are three
cases of the lucky accident
646
00:42:08 --> 00:42:14
which you should know.
The three cases,
647
00:42:12 --> 00:42:18
I feel justified calling it
"the" three cases.
648
00:42:17 --> 00:42:23
Oh, well, maybe I shouldn't do
that.
649
00:42:21 --> 00:42:27
The three most significant
examples are,
650
00:42:26 --> 00:42:32
example number one,
when A is a constant times the
651
00:42:31 --> 00:42:37
identity matrix.
In other words,
652
00:42:36 --> 00:42:42
when A is a matrix that looks
like this.
653
00:42:39 --> 00:42:45
That matrix commutes with every
other square matrix.
654
00:42:43 --> 00:42:49
If that is A,
then this law is always true
655
00:42:46 --> 00:42:52
and you are allowed to use this.
Okay, so that is one case.
656
00:42:51 --> 00:42:57
Another case,
when A is more general,
657
00:42:54 --> 00:43:00
is when B is equal to negative
A.
658
00:42:59 --> 00:43:05
I think you can see that that
is going to work because A times
659
00:43:03 --> 00:43:09
minus A is equal to minus A
times A.
660
00:43:07 --> 00:43:13
Yeah, they are both equal to A
squared,
661
00:43:11 --> 00:43:17
except with a negative sign in
front.
662
00:43:14 --> 00:43:20
And the third case is when B is
equal to the inverse of A
663
00:43:18 --> 00:43:24
because A A inverse is the same
as A inverse A.
664
00:43:23 --> 00:43:29
They are both the identity.
665
00:43:26 --> 00:43:32
Of course, A must have an
inverse.
666
00:43:30 --> 00:43:36
Okay, let's suppose it does.
Now, of them this is,
667
00:43:34 --> 00:43:40
I think, the most important one
because it leads to this law.
668
00:43:40 --> 00:43:46
That is forbidden,
but there is one case of it
669
00:43:44 --> 00:43:50
which is not forbidden and that
is here.
670
00:43:48 --> 00:43:54
What will it say?
Well, that will say that e to
671
00:43:52 --> 00:43:58
the A minus A is equal to e to
the A times e to
672
00:43:58 --> 00:44:04
the negative A.
This is true,
673
00:44:03 --> 00:44:09
even though the general law is
false.
674
00:44:05 --> 00:44:11
That is because A and negative
A commute with each other.
675
00:44:10 --> 00:44:16
But now what does this say?
What is e to the zero matrix?
676
00:44:14 --> 00:44:20
In other words,
suppose I take the matrix that
677
00:44:18 --> 00:44:24
is zero and plug it into the
formula for e?
678
00:44:21 --> 00:44:27
What do you get?
e to the zero times t is I.
679
00:44:24 --> 00:44:30
It has to be a two-by-two
matrix if it is going to be
680
00:44:29 --> 00:44:35
anything.
It is the matrix I.
681
00:44:33 --> 00:44:39
This side is I.
This side is the exponential
682
00:44:38 --> 00:44:44
matrix.
And what does that show?
683
00:44:41 --> 00:44:47
It shows that the inverse
matrix, the e to the A,
684
00:44:47 --> 00:44:53
is e to the negative A.
That is a very useful fact.
685
00:44:53 --> 00:44:59
This is the main survivor of
the exponential law.
686
00:45:00 --> 00:45:06
In general it is false,
but this standard corollary to
687
00:45:05 --> 00:45:11
the exponential law is true,
is equal to e to the minus A,
688
00:45:10 --> 00:45:16
just what you would
dream and hope would be true.
689
00:45:16 --> 00:45:22
Okay.
I have exactly two and a half
690
00:45:19 --> 00:45:25
minutes left in which to do the
impossible.
691
00:45:23 --> 00:45:29
All right.
The question is,
692
00:45:25 --> 00:45:31
how do you calculate e to the
At?
693
00:45:31 --> 00:45:37
You could use series,
but it rarely works.
694
00:45:34 --> 00:45:40
It is too hard.
There are a few examples,
695
00:45:38 --> 00:45:44
and you will have some more for
homework, but in general it is
696
00:45:43 --> 00:45:49
too hard because it is too hard
to calculate the powers of a
697
00:45:49 --> 00:45:55
general matrix A.
There is another method,
698
00:45:52 --> 00:45:58
which is useful only for
matrices which are symmetric,
699
00:45:57 --> 00:46:03
but like that --
Well, it is more than
700
00:46:01 --> 00:46:07
symmetric.
These two have to be the same.
701
00:46:04 --> 00:46:10
But you can handle those,
as you will see from the
702
00:46:07 --> 00:46:13
homework problems,
by breaking it up this way and
703
00:46:11 --> 00:46:17
using the exponential law.
This would be zero,
704
00:46:14 --> 00:46:20
b, b, zero.
705
00:46:16 --> 00:46:22
See, these two matrices commute
with each other and,
706
00:46:19 --> 00:46:25
therefore, I could use the
exponential law.
707
00:46:22 --> 00:46:28
This leaves all other cases.
And here is the way to handle
708
00:46:26 --> 00:46:32
all other cases.
All other cases.
709
00:46:30 --> 00:46:36
In other words,
if you cannot calculate the
710
00:46:33 --> 00:46:39
series, this trick doesn't work,
I have done as follows.
711
00:46:38 --> 00:46:44
You start with an arbitrary
fundamental matrix,
712
00:46:41 --> 00:46:47
not the exponential matrix.
You multiply it by its value at
713
00:46:46 --> 00:46:52
zero, that is a constant matrix,
and you take the inverse of
714
00:46:51 --> 00:46:57
that constant matrix.
It will have one because,
715
00:46:55 --> 00:47:01
remember, the fundamental
matrix never has the determinant
716
00:47:00 --> 00:47:06
zero.
So you can always take its
717
00:47:04 --> 00:47:10
inverse-ready value of t.
Now, what property does this
718
00:47:09 --> 00:47:15
have?
It is a fundamental matrix.
719
00:47:12 --> 00:47:18
How do I know that?
Well, because I found all
720
00:47:16 --> 00:47:22
fundamental matrices for you.
Take any one,
721
00:47:21 --> 00:47:27
multiply it by a square matrix
on the right-hand side,
722
00:47:26 --> 00:47:32
and you get still a fundamental
matrix.
723
00:47:29 --> 00:47:35
And what is its value at zero?
Well, it is x of zero times x
724
00:47:37 --> 00:47:43
of zero inverse.
Its value at zero is the
725
00:47:42 --> 00:47:48
identity.
Now, e to the At has
726
00:47:48 --> 00:47:54
these same two properties.
727
00:47:52 --> 00:47:58
728
00:47:56 --> 00:48:02
Namely, it is a fundamental
matrix and its value at zero is
729
00:48:01 --> 00:48:07
the identity. Conclusion,
this is e to the At.
730
00:48:05 --> 00:48:11
And that is the garden variety
731
00:48:08 --> 00:48:14
method of calculating the
exponential matrix,
732
00:48:11 --> 00:48:17
if you want to give it
explicitly.
733
00:48:13 --> 00:48:19
Start with any fundamental
matrix calculated,
734
00:48:16 --> 00:48:22
you should forgive the
expression using eigenvalues and
735
00:48:20 --> 00:48:26
eigenvectors and putting the
solutions into the columns.
736
00:48:24 --> 00:48:30
Evaluate it at zero,
take its inverse and multiply
737
00:48:28 --> 00:48:34
the two.
And what you end up with has to
738
00:48:32 --> 00:48:38
be the same as the thing
calculated with that infinite
739
00:48:36 --> 00:48:42
series.
Okay.
740
00:48:36 --> 00:48:42
You will get lots of practice
for homework and tomorrow.