WEBVTT
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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.
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,
00:13:08.000 --> 00:13:14.000
for the moment,
clueless.
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If you are clueless the place
to look always is do I know
00:13:14.000 --> 00:13:20.000
anything about this sort of
thing?
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I mean is there some special
case of this problem I can solve
00:13:22.000 --> 00:13:28.000
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.
00:14:20.000 --> 00:14:26.000
Well, in that case,
I am trying to solve the
00:14:23.000 --> 00:14:29.000
system.
The system consists of a single
00:14:26.000 --> 00:14:32.000
equation.
That is the way the system
00:14:29.000 --> 00:14:35.000
looks.
How do you solve that?
00:14:33.000 --> 00:14:39.000
Well, you were born knowing how
to solve that.
00:14:37.000 --> 00:14:43.000
Anyway, you certainly didn't
learn it in this course.
00:14:42.000 --> 00:14:48.000
You separate variables,
blah, blah, blah,
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and the solution is x equals,
the basic solution is e to the
00:14:52.000 --> 00:14:58.000
at,
and you multiply that by an
00:14:56.000 --> 00:15:02.000
arbitrary constant.
Now, that is a formula for the
00:15:02.000 --> 00:15:08.000
solution.
And it uses the parameter in
00:15:05.000 --> 00:15:11.000
the equation.
I didn't have to know a special
00:15:09.000 --> 00:15:15.000
number.
I didn't have to put a
00:15:11.000 --> 00:15:17.000
particular number here to use
that.
00:15:14.000 --> 00:15:20.000
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.
00:15:23.000 --> 00:15:29.000
But let's take a quick look as
to why this works.
00:15:29.000 --> 00:15:35.000
Of course, you separate
variables and use calculus.
00:15:32.000 --> 00:15:38.000
I am going to give you a
slightly different argument that
00:15:36.000 --> 00:15:42.000
has the advantage of
generalizing to the end-by-end
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case.
And the argument goes as
00:15:41.000 --> 00:15:47.000
follows for that.
It uses the definition of the
00:15:44.000 --> 00:15:50.000
exponential function not as the
inverse to the logarithm,
00:15:48.000 --> 00:15:54.000
which is where the fancy
calculus books get it from,
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nor as the naďve high school
method, e squared means
00:15:56.000 --> 00:16:02.000
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.
00:16:07.000 --> 00:16:13.000
So, the naďve definition of the
exponential function.
00:16:11.000 --> 00:16:17.000
Instead, I will use the
definition of the exponential
00:16:14.000 --> 00:16:20.000
function that comes from an
infinite series.
00:16:17.000 --> 00:16:23.000
Leaving out the arbitrary
constant that we don't have to
00:16:21.000 --> 00:16:27.000
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.
00:16:32.000 --> 00:16:38.000
I will put out one more term
and let's call it quits there.
00:16:38.000 --> 00:16:44.000
If I take this then argument
goes let's just differentiate
00:16:44.000 --> 00:16:50.000
it.
In other words,
00:16:46.000 --> 00:16:52.000
what is the derivative of e to
the at with respect to t?
00:16:52.000 --> 00:16:58.000
Well, just differentiating term
00:16:57.000 --> 00:17:03.000
by term it is zero plus the
first term is a,
00:17:01.000 --> 00:17:07.000
the next term is a squared
times t.
00:17:08.000 --> 00:17:14.000
This differentiates to t
squared over two factorial.
00:17:13.000 --> 00:17:19.000
And the answer is that this is
00:17:17.000 --> 00:17:23.000
equal to a times,
if you factor out the a,
00:17:21.000 --> 00:17:27.000
what is left is one plus a t
plus a squared t squared over
00:17:26.000 --> 00:17:32.000
two factorial
00:17:32.000 --> 00:17:38.000
In other words,
it is simply e to the at.
00:17:34.000 --> 00:17:40.000
In other words,
00:17:36.000 --> 00:17:42.000
by differentiating the series,
using the series definition of
00:17:40.000 --> 00:17:46.000
the exponential and by
differentiating it term by term,
00:17:44.000 --> 00:17:50.000
I can immediately see that is
satisfies this differential
00:17:48.000 --> 00:17:54.000
equation.
What about the arbitrary
00:17:50.000 --> 00:17:56.000
constant?
Well, if you would like,
00:17:52.000 --> 00:17:58.000
you can include it here,
but it is easier to observe
00:17:56.000 --> 00:18:02.000
that by linearity if e to the a
t solves the equation so does
00:18:00.000 --> 00:18:06.000
the constant times it because
the equation is linear.
00:18:05.000 --> 00:18:11.000
Now, that is the idea that I am
going to use to solve the system
00:18:12.000 --> 00:18:18.000
in general.
What are we doing to say?
00:18:16.000 --> 00:18:22.000
Well, what could we say?
The solution to,
00:18:21.000 --> 00:18:27.000
well, let's get two solutions
at once by writing a fundamental
00:18:28.000 --> 00:18:34.000
matrix.
"A" fundamental matrix,
00:18:33.000 --> 00:18:39.000
I don't claim it is "the" one,
for the system x prime equals A
00:18:41.000 --> 00:18:47.000
x.
That is what we are trying to
00:18:46.000 --> 00:18:52.000
solve.
And we are going to get two
00:18:51.000 --> 00:18:57.000
solutions by getting a
fundamental matrix for it.
00:18:57.000 --> 00:19:03.000
The answer is e to the a t.
00:19:04.000 --> 00:19:10.000
Isn't that what it should be?
I had a little a.
00:19:07.000 --> 00:19:13.000
Now we have a matrix.
Okay, just put the matrix up
00:19:11.000 --> 00:19:17.000
there.
Now, what on earth?
00:19:13.000 --> 00:19:19.000
The first person who must have
thought of this,
00:19:16.000 --> 00:19:22.000
it happened about 100 years
ago, what meaning should be
00:19:20.000 --> 00:19:26.000
given to e to a matrix power?
Well, clearly the two na•ve
00:19:25.000 --> 00:19:31.000
definitions won't work.
The only possible meaning you
00:19:30.000 --> 00:19:36.000
could try for is using the
infinite series,
00:19:33.000 --> 00:19:39.000
but that does work.
So this is a definition I am
00:19:37.000 --> 00:19:43.000
giving you, the exponential
matrix.
00:19:40.000 --> 00:19:46.000
Now, notice the A is a
two-by-two matrix multiplying it
00:19:44.000 --> 00:19:50.000
by t.
What I have up here is that
00:19:46.000 --> 00:19:52.000
it's basically a two-by-two
matrix.
00:19:49.000 --> 00:19:55.000
Its entries involve t,
but it's a two-by-two matrix.
00:19:53.000 --> 00:19:59.000
Okay.
We are trying to get the analog
00:19:56.000 --> 00:20:02.000
of that formula over there.
Well, leave the first term out
00:20:02.000 --> 00:20:08.000
just for a moment.
The next term is going to
00:20:05.000 --> 00:20:11.000
surely be A times t.
This is a two-by-two matrix,
00:20:09.000 --> 00:20:15.000
right?
What should the next term be?
00:20:12.000 --> 00:20:18.000
Well, A squared times t squared
over two factorial.
00:20:16.000 --> 00:20:22.000
What kind of a guy is that?
Well, if A is a two-by-two
00:20:20.000 --> 00:20:26.000
matrix so is A squared.
How about this?
00:20:23.000 --> 00:20:29.000
This is just a scalar which
multiplies every entry of A
00:20:28.000 --> 00:20:34.000
squared.
And, therefore,
00:20:30.000 --> 00:20:36.000
this is still a two-by-two
matrix.
00:20:33.000 --> 00:20:39.000
That is a two-by-two matrix.
This is a two-by-two matrix.
00:20:36.000 --> 00:20:42.000
No matter how many times you
multiply A by itself it stays a
00:20:40.000 --> 00:20:46.000
two-by-two matrix.
It gets more and more
00:20:43.000 --> 00:20:49.000
complicated looking but it is
always a two-by-two matrix.
00:20:46.000 --> 00:20:52.000
And now I am multiplying every
entry of that by the scalar t
00:20:50.000 --> 00:20:56.000
cubed over three factorial.
00:20:53.000 --> 00:20:59.000
I am continuing on in that way.
What I get, therefore,
00:20:56.000 --> 00:21:02.000
is a sum of two-by-two
matrices.
00:21:00.000 --> 00:21:06.000
Well, you can add two-by-two
matrices to each other.
00:21:03.000 --> 00:21:09.000
We've never made an infinite
series of them,
00:21:06.000 --> 00:21:12.000
we haven't done it,
but others have.
00:21:09.000 --> 00:21:15.000
And this is what they wrote.
The only question is,
00:21:12.000 --> 00:21:18.000
what should we put in the
beginning?
00:21:15.000 --> 00:21:21.000
Over there I have the number
one.
00:21:17.000 --> 00:21:23.000
But I, of course,
cannot add the number one to a
00:21:20.000 --> 00:21:26.000
two-by-two matrices.
What goes here must be a
00:21:23.000 --> 00:21:29.000
two-by-two matrix,
which is the closest thing to
00:21:27.000 --> 00:21:33.000
one I can think of.
What should it be?
00:21:31.000 --> 00:21:37.000
The I.
Two-by-two I.
00:21:32.000 --> 00:21:38.000
Two-by-two identity matrix
looks like the natural candidate
00:21:37.000 --> 00:21:43.000
for what to put there.
And, in fact,
00:21:40.000 --> 00:21:46.000
it is the right thing to put
there.
00:21:42.000 --> 00:21:48.000
Okay.
Now I have a conjecture,
00:21:45.000 --> 00:21:51.000
you know, purely formally,
changing only with a keystroke
00:21:49.000 --> 00:21:55.000
of the computer,
all the little a's have been
00:21:53.000 --> 00:21:59.000
changed to capital A's.
And now all I have to do is
00:21:57.000 --> 00:22:03.000
wonder if this is going to work.
Well, what is the basic thing I
00:22:03.000 --> 00:22:09.000
have to check to see that it is
the fundamental matrix?
00:22:07.000 --> 00:22:13.000
The question is,
I wrote it down all right,
00:22:11.000 --> 00:22:17.000
but is this a fundamental
matrix for the system?
00:22:14.000 --> 00:22:20.000
Well, I have a way of
recognizing a fundamental matrix
00:22:19.000 --> 00:22:25.000
when I see one.
The critical thing is that it
00:22:22.000 --> 00:22:28.000
should satisfy this matrix
differential equation.
00:22:26.000 --> 00:22:32.000
That is what I should verify.
Does this guy that I have
00:22:32.000 --> 00:22:38.000
written down satisfy this
equation?
00:22:35.000 --> 00:22:41.000
And the answer is,
number two is,
00:22:37.000 --> 00:22:43.000
it satisfies x prime equals Ax.
00:22:41.000 --> 00:22:47.000
In other words,
plugging in x equals this e to
00:22:45.000 --> 00:22:51.000
the at,
whose definition I just gave
00:22:49.000 --> 00:22:55.000
you.
If I substitute that in,
00:22:51.000 --> 00:22:57.000
does it satisfy that matrix
differential equation?
00:22:56.000 --> 00:23:02.000
The answer is yes.
I am not going to calculate it
00:23:00.000 --> 00:23:06.000
out because the calculation is
actually identical to what I did
00:23:04.000 --> 00:23:10.000
there.
The only difference is when I
00:23:06.000 --> 00:23:12.000
differentiated it term by term,
how do you differentiate
00:23:10.000 --> 00:23:16.000
something like this?
Well, you differentiate every
00:23:13.000 --> 00:23:19.000
term in it.
But, if you work it out,
00:23:15.000 --> 00:23:21.000
this is a constant matrix,
every term of which is
00:23:18.000 --> 00:23:24.000
multiplied by t squared over two
factorial.
00:23:21.000 --> 00:23:27.000
Well, if you differentiate
every entry of that constant,
00:23:25.000 --> 00:23:31.000
of that matrix,
what you are going to get is A
00:23:27.000 --> 00:23:33.000
squared times just the
derivative of that part,
00:23:30.000 --> 00:23:36.000
which is simply t.
In other words,
00:23:34.000 --> 00:23:40.000
the formal calculation looks
absolutely identical to that.
00:23:40.000 --> 00:23:46.000
So the answer to this is yes,
by the same calculation as
00:23:45.000 --> 00:23:51.000
before, as for the one-by-one
case.
00:23:48.000 --> 00:23:54.000
And now the only other thing to
check is that the determinant is
00:23:55.000 --> 00:24:01.000
not zero.
In fact, the determinant is not
00:23:59.000 --> 00:24:05.000
zero at one point.
That is all you have to check.
00:24:04.000 --> 00:24:10.000
What is x of zero?
What is the value of the
00:24:08.000 --> 00:24:14.000
determinant of x is e
to the At?
00:24:11.000 --> 00:24:17.000
What is the value of this thing
at zero?
00:24:14.000 --> 00:24:20.000
Here is my function.
If I plug in t equals zero,
00:24:18.000 --> 00:24:24.000
what is it equal to?
I.
00:24:20.000 --> 00:24:26.000
What is the determinant of I?
One.
00:24:22.000 --> 00:24:28.000
It is certainly not zero.
00:24:38.000 --> 00:24:44.000
By writing down this infinite
series, I have my two solutions.
00:24:41.000 --> 00:24:47.000
Its columns give me two
solutions to the original
00:24:44.000 --> 00:24:50.000
system.
There were no eigenvalues,
00:24:47.000 --> 00:24:53.000
no eigenvectors.
I have a formula for the
00:24:49.000 --> 00:24:55.000
answer.
What is the formula?
00:24:51.000 --> 00:24:57.000
It is e to the At.
And, of course,
00:24:54.000 --> 00:25:00.000
anybody reading the paper is
supposed to know what e to the
00:24:57.000 --> 00:25:03.000
At is.
It means that.
00:25:00.000 --> 00:25:06.000
This is just marvelous.
There must be a fly in the
00:25:03.000 --> 00:25:09.000
ointment somewhere.
Only a teeny little fly.
00:25:07.000 --> 00:25:13.000
There is a teeny little fly
because it is almost impossible
00:25:11.000 --> 00:25:17.000
to calculate that series for all
reasonable times.
00:25:15.000 --> 00:25:21.000
However, once in a while it is.
Let me give you an example
00:25:20.000 --> 00:25:26.000
where it is possible to
calculate the series and were
00:25:24.000 --> 00:25:30.000
you get a nice answer.
Let's work out an example.
00:25:36.000 --> 00:25:42.000
By the way, you know,
nowadays, we are not back 50
00:25:40.000 --> 00:25:46.000
years, the exponential matrix
has the same status on,
00:25:45.000 --> 00:25:51.000
say, a Matlab or Maple or
Mathematica, as the ordinary
00:25:51.000 --> 00:25:57.000
exponential function does.
It is just a command you type
00:25:56.000 --> 00:26:02.000
in.
You type in your matrix.
00:26:00.000 --> 00:26:06.000
And you now say EXP of that
matrix and out comes the answer
00:26:04.000 --> 00:26:10.000
to as many decimal places as you
want.
00:26:06.000 --> 00:26:12.000
It will be square matrix with
entries carefully written out.
00:26:10.000 --> 00:26:16.000
So, in that sense,
the fact that we cannot
00:26:13.000 --> 00:26:19.000
calculate it shouldn't bother
us.
00:26:15.000 --> 00:26:21.000
There are machines to do the
calculations.
00:26:18.000 --> 00:26:24.000
What we are interested in is it
as a theoretical tool.
00:26:22.000 --> 00:26:28.000
But, in order to get any
feeling for this at all,
00:26:25.000 --> 00:26:31.000
we certainly have to do a few
calculations.
00:26:30.000 --> 00:26:36.000
Let's do an easy one.
Let's consider the system x
00:26:34.000 --> 00:26:40.000
prime equals y,
y prime equals x.
00:26:39.000 --> 00:26:45.000
This is very easily done by
elimination, but that is
00:26:43.000 --> 00:26:49.000
forbidden.
First of all,
00:26:45.000 --> 00:26:51.000
we write it as a matrix.
It's the system x prime equals
00:26:50.000 --> 00:26:56.000
zero, one, one,
zero, x.
00:26:53.000 --> 00:26:59.000
Here is my A.
00:26:55.000 --> 00:27:01.000
And so e to the At
is going to be --
00:27:01.000 --> 00:27:07.000
A is zero, one,
one, zero.
00:27:02.000 --> 00:27:08.000
What we want to
00:27:05.000 --> 00:27:11.000
calculate is we are going to get
both solutions at once by
00:27:08.000 --> 00:27:14.000
calculating it one fell swoop e
to the At.
00:27:12.000 --> 00:27:18.000
Okay.
E to the At equals.
00:27:13.000 --> 00:27:19.000
I am going to actually write
out these guys.
00:27:16.000 --> 00:27:22.000
Well, obviously the hard part,
the part which is normally
00:27:20.000 --> 00:27:26.000
going to prevent us from
calculating this series
00:27:23.000 --> 00:27:29.000
explicitly, by hand anyway,
because, as I said,
00:27:26.000 --> 00:27:32.000
the computer can always do it.
The value, how do we raise a
00:27:32.000 --> 00:27:38.000
matrix to a high power?
You just keep multiplying and
00:27:37.000 --> 00:27:43.000
multiplying and multiplying.
That looks like a rather
00:27:41.000 --> 00:27:47.000
forbidding and unpromising
activity.
00:27:44.000 --> 00:27:50.000
Well, here it is easy.
Let's see what happens.
00:27:48.000 --> 00:27:54.000
If that is A,
what is A squared?
00:27:51.000 --> 00:27:57.000
I am going to have to calculate
that as part of the series.
00:27:56.000 --> 00:28:02.000
That is going to be zero,
one, one, zero times zero,
00:28:00.000 --> 00:28:06.000
one, one, zero,
which is one,
00:28:03.000 --> 00:28:09.000
zero, zero, one.
00:28:10.000 --> 00:28:16.000
We got saved.
It is the identity.
00:28:13.000 --> 00:28:19.000
Now, from this point on we
don't have to do anymore
00:28:17.000 --> 00:28:23.000
calculations,
but I will do them anyway.
00:28:21.000 --> 00:28:27.000
What is A cubed?
Don't start from scratch again.
00:28:26.000 --> 00:28:32.000
No, no, no.
A cubed is A squared times A.
00:28:30.000 --> 00:28:36.000
And A squared is,
00:28:34.000 --> 00:28:40.000
in real life,
the identity.
00:28:35.000 --> 00:28:41.000
Of course, you would do all
this in your head,
00:28:38.000 --> 00:28:44.000
but I am being a good boy and
writing it all out.
00:28:41.000 --> 00:28:47.000
This is I, the identity,
times A, which is A.
00:28:44.000 --> 00:28:50.000
I will do one more.
What is A to the fourth?
00:28:47.000 --> 00:28:53.000
Now, you would be tempted to
say A to the fourth is A
00:28:50.000 --> 00:28:56.000
squared, which is I times I,
which is I, but that would be
00:28:54.000 --> 00:29:00.000
wrong.
A to the fourth is A cubed
00:28:58.000 --> 00:29:04.000
times A,
which is, I have just
00:29:02.000 --> 00:29:08.000
calculated is A times A,
right?
00:29:05.000 --> 00:29:11.000
And now that is A squared,
which is the identity.
00:29:10.000 --> 00:29:16.000
It is clear,
by this argument,
00:29:12.000 --> 00:29:18.000
it is going to continue in the
same way each time you add an A
00:29:18.000 --> 00:29:24.000
on the right-hand side,
you are going to keep
00:29:22.000 --> 00:29:28.000
alternating between the
identity, A, the next one will
00:29:27.000 --> 00:29:33.000
be identity, the next will be A.
The end result is that the
00:29:34.000 --> 00:29:40.000
first term of the series is
simply the identity;
00:29:39.000 --> 00:29:45.000
the next term of the series is
A, but it is multiplied by t.
00:29:44.000 --> 00:29:50.000
I will keep the t on the
outside.
00:29:47.000 --> 00:29:53.000
Remember, when you multiply a
matrix by a scalar,
00:29:52.000 --> 00:29:58.000
that means multiply every entry
by that scalar.
00:29:56.000 --> 00:30:02.000
This is the matrix zero,
t, t, zero.
00:30:00.000 --> 00:30:06.000
I will do a couple more terms.
00:30:05.000 --> 00:30:11.000
The next term would be,
well, A squared we just
00:30:08.000 --> 00:30:14.000
calculated as the identity.
That looks like this.
00:30:12.000 --> 00:30:18.000
Except now I multiply every
term by t squared over two
00:30:16.000 --> 00:30:22.000
factorial.
All right.
00:30:19.000 --> 00:30:25.000
I'll go for broke.
The next one will be this times
00:30:22.000 --> 00:30:28.000
t cubed over three factorial.
00:30:25.000 --> 00:30:31.000
And, fortunately,
I have run out of room.
00:30:30.000 --> 00:30:36.000
Okay, let's calculate then.
00:30:54.000 --> 00:31:00.000
What is the final answer for e
to At?
00:30:57.000 --> 00:31:03.000
I have an infinite series of
two-by-two matrices.
00:31:00.000 --> 00:31:06.000
Let's look at the term in the
upper left-hand corner.
00:31:03.000 --> 00:31:09.000
It is one plus zero times t
plus one times t squared over
00:31:07.000 --> 00:31:13.000
two factorial plus zero
times t.
00:31:11.000 --> 00:31:17.000
It is going to be,
00:31:12.000 --> 00:31:18.000
in other words,
one plus t squared over two
00:31:15.000 --> 00:31:21.000
factorial plus
the next term,
00:31:18.000 --> 00:31:24.000
which is not on the board but I
think you can see,
00:31:21.000 --> 00:31:27.000
is this.
And it continues on in the same
00:31:24.000 --> 00:31:30.000
way.
How about the lower left term?
00:31:28.000 --> 00:31:34.000
Well, that is zero plus t plus
zero plus t cubed over three
00:31:32.000 --> 00:31:38.000
factorial and so on.
00:31:35.000 --> 00:31:41.000
It is t plus t cubed over three
factorial plus t to the fifth
00:31:39.000 --> 00:31:45.000
over five factorial.
00:31:42.000 --> 00:31:48.000
And the other terms in the
00:31:44.000 --> 00:31:50.000
other two corners are just the
same as these.
00:31:47.000 --> 00:31:53.000
This one, for example,
is zero plus t plus zero plus t
00:31:51.000 --> 00:31:57.000
cubed over three factorial.
00:31:55.000 --> 00:32:01.000
And the lower one is one plus
zero plus t squared
00:31:59.000 --> 00:32:05.000
and so on.
This is the same as one plus t
00:32:04.000 --> 00:32:10.000
squared over two factorial
and so on,
00:32:08.000 --> 00:32:14.000
and up here we have t plus t
cubed over three factorial
00:32:14.000 --> 00:32:20.000
and so on.
Well, that matrix doesn't look
00:32:18.000 --> 00:32:24.000
very square, but it is.
It is infinitely long
00:32:22.000 --> 00:32:28.000
physically, but it has one term
here, one term here,
00:32:27.000 --> 00:32:33.000
one term here and one term
there.
00:32:31.000 --> 00:32:37.000
Now, all we have to do is make
those terms look a little
00:32:35.000 --> 00:32:41.000
better.
For here I have to rely on the
00:32:38.000 --> 00:32:44.000
culture, which you may or may
not posses.
00:32:42.000 --> 00:32:48.000
You would know what these
series were if only they
00:32:46.000 --> 00:32:52.000
alternated their signs.
If this were a negative,
00:32:50.000 --> 00:32:56.000
negative, negative then the top
would be cosine t and
00:32:55.000 --> 00:33:01.000
this would be sine t,
but they don't.
00:33:01.000 --> 00:33:07.000
So they are the next best
thing.
00:33:04.000 --> 00:33:10.000
They are what?
Hyperbolic.
00:33:06.000 --> 00:33:12.000
The topic is not cosine t,
but cosh t.
00:33:11.000 --> 00:33:17.000
The bottle is sinh t.
00:33:15.000 --> 00:33:21.000
And how do we know this?
Because you remember.
00:33:19.000 --> 00:33:25.000
And what if I don't remember?
Well, you know now.
00:33:24.000 --> 00:33:30.000
That is why you come to class.
00:33:35.000 --> 00:33:41.000
Well, for those of you who
don't, remember,
00:33:38.000 --> 00:33:44.000
this is e to the t plus e to
the negative t.
00:33:44.000 --> 00:33:50.000
It should be over two,
but I don't have room to put in
00:33:48.000 --> 00:33:54.000
the two.
This doesn't mean I will omit
00:33:52.000 --> 00:33:58.000
it.
It just means I will put it in
00:33:55.000 --> 00:34:01.000
at the end by multiplying every
entry of this matrix by
00:34:00.000 --> 00:34:06.000
one-half.
If you have forgotten what cosh
00:34:04.000 --> 00:34:10.000
t is, it's e to the t plus e to
the negative t divided by two.
00:34:12.000 --> 00:34:18.000
And the similar thing for sinh
t.
00:34:14.000 --> 00:34:20.000
There is your first explicit
exponential matrix calculated
00:34:19.000 --> 00:34:25.000
according to the definition.
And what we have found is the
00:34:24.000 --> 00:34:30.000
solution to the system x prime
equals y,
00:34:28.000 --> 00:34:34.000
y prime equals x.
A fundamental matrix.
00:34:33.000 --> 00:34:39.000
In other words,
cosh t and sinh t satisfy both
00:34:36.000 --> 00:34:42.000
solutions to that system.
Now, there is one thing people
00:34:40.000 --> 00:34:46.000
love the exponential matrix in
particular for,
00:34:44.000 --> 00:34:50.000
and that is the ease with which
it solves the initial value
00:34:48.000 --> 00:34:54.000
problem.
It is exactly what happens when
00:34:51.000 --> 00:34:57.000
studying the single system,
the single equation x prime
00:34:55.000 --> 00:35:01.000
equals Ax,
but let's do it in general.
00:35:00.000 --> 00:35:06.000
Let's do it in general.
What is the initial value
00:35:03.000 --> 00:35:09.000
problem?
Well, the initial value problem
00:35:07.000 --> 00:35:13.000
is we start with our old system,
but now I want to plug in
00:35:11.000 --> 00:35:17.000
initial conditions.
I want the particular solution
00:35:15.000 --> 00:35:21.000
which satisfies the initial
condition.
00:35:18.000 --> 00:35:24.000
Let's make it zero to avoid
complications,
00:35:22.000 --> 00:35:28.000
to avoid a lot of notation.
This is to be some starting
00:35:26.000 --> 00:35:32.000
value.
This is a certain constant
00:35:29.000 --> 00:35:35.000
vector.
It is to be the value of the
00:35:33.000 --> 00:35:39.000
solution at zero.
And the problem is find what x
00:35:37.000 --> 00:35:43.000
of t is.
Well, if you are using the
00:35:41.000 --> 00:35:47.000
exponential matrix it is a joke.
It is a joke.
00:35:45.000 --> 00:35:51.000
Shall I derive it or just do
it?
00:35:48.000 --> 00:35:54.000
All right.
The general solution,
00:35:51.000 --> 00:35:57.000
let's derive it,
and then I will put up the
00:35:55.000 --> 00:36:01.000
final formula in a box so that
you will know it is important.
00:36:02.000 --> 00:36:08.000
What is the general solution?
Well, I did that for you at the
00:36:06.000 --> 00:36:12.000
beginning of the period.
Once you have a fundamental
00:36:09.000 --> 00:36:15.000
matrix, you get the general
solution by multiplying it on
00:36:13.000 --> 00:36:19.000
the right by an arbitrary
constant vector.
00:36:16.000 --> 00:36:22.000
The general solution is going
to be x equals e to the At.
00:36:20.000 --> 00:36:26.000
That is my super fundamental
00:36:22.000 --> 00:36:28.000
matrix, found without
eigenvalues and eigenvectors.
00:36:27.000 --> 00:36:33.000
And this should be multiplied
by some unknown constant vector
00:36:32.000 --> 00:36:38.000
c.
The only question is,
00:36:35.000 --> 00:36:41.000
what should the constant vector
c be?
00:36:38.000 --> 00:36:44.000
To find c, I will plug in zero.
When t is zero,
00:36:42.000 --> 00:36:48.000
here I get x of zero,
here I get e to the A times
00:36:47.000 --> 00:36:53.000
zero times c.
Now what is this?
00:36:51.000 --> 00:36:57.000
This is the vector of initial
conditions?
00:36:55.000 --> 00:37:01.000
What is e to the A times zero?
Plug in t equals zero.
00:37:00.000 --> 00:37:06.000
What do you get?
I.
00:37:04.000 --> 00:37:10.000
Therefore, c is what?
c is x zero.
00:37:11.000 --> 00:37:17.000
It is a total joke.
And the solution is,
00:37:17.000 --> 00:37:23.000
the initial value problem is x
equals e to the At
00:37:26.000 --> 00:37:32.000
times x zero.
It is just what it would have
00:37:32.000 --> 00:37:38.000
been at one variable.
The only difference is that
00:37:36.000 --> 00:37:42.000
here we are allowed to put the c
out front.
00:37:39.000 --> 00:37:45.000
In other words,
if I asked you to put in the
00:37:41.000 --> 00:37:47.000
initial condition,
you would probably write x
00:37:44.000 --> 00:37:50.000
equals little x zero times e to
the At.
00:37:48.000 --> 00:37:54.000
And you would be tempted to do
the same thing here,
00:37:52.000 --> 00:37:58.000
vector x equals vector x zero
times e to the At.
00:37:55.000 --> 00:38:01.000
Now, you cannot do that.
And, if you try to Matlab will
00:38:00.000 --> 00:38:06.000
hiccup and say illegal
operation.
00:38:02.000 --> 00:38:08.000
What is the illegal operation?
Well, x is a column vector.
00:38:07.000 --> 00:38:13.000
From the system it is a column
vector.
00:38:10.000 --> 00:38:16.000
That means the initial
conditions are also a column
00:38:14.000 --> 00:38:20.000
vector.
You cannot multiply a column
00:38:17.000 --> 00:38:23.000
vector out front and a square
matrix afterwards.
00:38:20.000 --> 00:38:26.000
You cannot.
If you want to multiply a
00:38:23.000 --> 00:38:29.000
matrix by a column vector,
it has to come afterwards so
00:38:27.000 --> 00:38:33.000
you can do zing,
zing.
00:38:31.000 --> 00:38:37.000
There is no zing,
you see.
00:38:33.000 --> 00:38:39.000
You cannot put it in front.
It doesn't work.
00:38:36.000 --> 00:38:42.000
So it must go behind.
That is the only place you
00:38:40.000 --> 00:38:46.000
might get tripped up.
And, as I say,
00:38:43.000 --> 00:38:49.000
if you try to type that in
using Matlab,
00:38:47.000 --> 00:38:53.000
you will immediately get error
messages that it is illegal,
00:38:52.000 --> 00:38:58.000
you cannot do that.
Anyway, we have our solution.
00:38:56.000 --> 00:39:02.000
There is our system.
Our initial value problem
00:39:00.000 --> 00:39:06.000
anyway is in pink,
and its solution using the
00:39:04.000 --> 00:39:10.000
exponential matrix is in green.
Now, the only problem is we
00:39:08.000 --> 00:39:14.000
still have to talk a little bit
more about calculating this.
00:39:13.000 --> 00:39:19.000
Now, the principle warning with
an exponential matrix is that
00:39:17.000 --> 00:39:23.000
once you have gotten by the
simplest things involving the
00:39:21.000 --> 00:39:27.000
fact that it solves systems,
it gives you the fundamental
00:39:26.000 --> 00:39:32.000
matrix for a system,
then you start flexing your
00:39:29.000 --> 00:39:35.000
muscles and say,
oh, well, let's see what else
00:39:33.000 --> 00:39:39.000
we can do with this.
For example,
00:39:36.000 --> 00:39:42.000
the reason exponentials came
into being in the first place
00:39:40.000 --> 00:39:46.000
was because of the exponential
law, right?
00:39:43.000 --> 00:39:49.000
I will kill anybody who sends
me emails saying,
00:39:46.000 --> 00:39:52.000
what is the exponential law?
The exponential law would say
00:39:50.000 --> 00:39:56.000
that e to the A plus B is equal
to e to the A times e to the B.
00:39:54.000 --> 00:40:00.000
The law of exponents,
00:39:58.000 --> 00:40:04.000
in other words.
It is the thing that makes the
00:40:01.000 --> 00:40:07.000
exponential function different
from all other functions that it
00:40:05.000 --> 00:40:11.000
satisfies something like that.
Now, first of all,
00:40:08.000 --> 00:40:14.000
does this make sense?
That is are the symbols
00:40:11.000 --> 00:40:17.000
compatible?
Let's see.
00:40:13.000 --> 00:40:19.000
This is a two-by-two matrix,
this is a two-by-two matrix,
00:40:16.000 --> 00:40:22.000
so it does make sense to
multiply them,
00:40:19.000 --> 00:40:25.000
and the answer will be a
two-by-two matrix.
00:40:21.000 --> 00:40:27.000
How about here?
This is a two-by-two matrix,
00:40:24.000 --> 00:40:30.000
add this to it.
It is still a two-by-two
00:40:27.000 --> 00:40:33.000
matrix.
e to a two-by-two matrix still
00:40:29.000 --> 00:40:35.000
comes out to be a two-by-two
matrix.
00:40:33.000 --> 00:40:39.000
Both sides are legitimate
two-by-two matrices.
00:40:37.000 --> 00:40:43.000
The only question is,
are they equal?
00:40:41.000 --> 00:40:47.000
And the answer is not in a
pig's eye.
00:40:45.000 --> 00:40:51.000
How could this be?
Well, I didn't make up these
00:40:50.000 --> 00:40:56.000
laws.
I just obey them.
00:40:52.000 --> 00:40:58.000
I wish I had time to do a
little calculation to show that
00:40:58.000 --> 00:41:04.000
it is not true.
It is true in certain special
00:41:03.000 --> 00:41:09.000
cases.
It is true in the special case,
00:41:06.000 --> 00:41:12.000
and this is pretty much if and
only if, the only case in which
00:41:12.000 --> 00:41:18.000
it is true is if A and B are not
arbitrary square matrices but
00:41:17.000 --> 00:41:23.000
commute with each other.
You see, if you start writing
00:41:22.000 --> 00:41:28.000
out the series to try to check
whether that law is true,
00:41:27.000 --> 00:41:33.000
you will get a bunch of terms
here, a bunch of terms here.
00:41:34.000 --> 00:41:40.000
And you will find that those
terms are pair-wise equal only
00:41:38.000 --> 00:41:44.000
if you are allowed to let the
matrices commute with each
00:41:41.000 --> 00:41:47.000
other.
In other words,
00:41:43.000 --> 00:41:49.000
if you can turn AB plus BA into
twice AB then
00:41:47.000 --> 00:41:53.000
everything will work fine.
But if you cannot do that it
00:41:51.000 --> 00:41:57.000
will not.
Now, when do two square
00:41:53.000 --> 00:41:59.000
matrices commute with each
other?
00:41:56.000 --> 00:42:02.000
The answer is almost never.
It is just a lucky accident if
00:42:02.000 --> 00:42:08.000
they do, but there are three
cases of the lucky accident
00:42:08.000 --> 00:42:14.000
which you should know.
The three cases,
00:42:12.000 --> 00:42:18.000
I feel justified calling it
"the" three cases.
00:42:17.000 --> 00:42:23.000
Oh, well, maybe I shouldn't do
that.
00:42:21.000 --> 00:42:27.000
The three most significant
examples are,
00:42:26.000 --> 00:42:32.000
example number one,
when A is a constant times the
00:42:31.000 --> 00:42:37.000
identity matrix.
In other words,
00:42:36.000 --> 00:42:42.000
when A is a matrix that looks
like this.
00:42:39.000 --> 00:42:45.000
That matrix commutes with every
other square matrix.
00:42:43.000 --> 00:42:49.000
If that is A,
then this law is always true
00:42:46.000 --> 00:42:52.000
and you are allowed to use this.
Okay, so that is one case.
00:42:51.000 --> 00:42:57.000
Another case,
when A is more general,
00:42:54.000 --> 00:43:00.000
is when B is equal to negative
A.
00:42:59.000 --> 00:43:05.000
I think you can see that that
is going to work because A times
00:43:03.000 --> 00:43:09.000
minus A is equal to minus A
times A.
00:43:07.000 --> 00:43:13.000
Yeah, they are both equal to A
squared,
00:43:11.000 --> 00:43:17.000
except with a negative sign in
front.
00:43:14.000 --> 00:43:20.000
And the third case is when B is
equal to the inverse of A
00:43:18.000 --> 00:43:24.000
because A A inverse is the same
as A inverse A.
00:43:23.000 --> 00:43:29.000
They are both the identity.
00:43:26.000 --> 00:43:32.000
Of course, A must have an
inverse.
00:43:30.000 --> 00:43:36.000
Okay, let's suppose it does.
Now, of them this is,
00:43:34.000 --> 00:43:40.000
I think, the most important one
because it leads to this law.
00:43:40.000 --> 00:43:46.000
That is forbidden,
but there is one case of it
00:43:44.000 --> 00:43:50.000
which is not forbidden and that
is here.
00:43:48.000 --> 00:43:54.000
What will it say?
Well, that will say that e to
00:43:52.000 --> 00:43:58.000
the A minus A is equal to e to
the A times e to
00:43:58.000 --> 00:44:04.000
the negative A.
This is true,
00:44:03.000 --> 00:44:09.000
even though the general law is
false.
00:44:05.000 --> 00:44:11.000
That is because A and negative
A commute with each other.
00:44:10.000 --> 00:44:16.000
But now what does this say?
What is e to the zero matrix?
00:44:14.000 --> 00:44:20.000
In other words,
suppose I take the matrix that
00:44:18.000 --> 00:44:24.000
is zero and plug it into the
formula for e?
00:44:21.000 --> 00:44:27.000
What do you get?
e to the zero times t is I.
00:44:24.000 --> 00:44:30.000
It has to be a two-by-two
matrix if it is going to be
00:44:29.000 --> 00:44:35.000
anything.
It is the matrix I.
00:44:33.000 --> 00:44:39.000
This side is I.
This side is the exponential
00:44:38.000 --> 00:44:44.000
matrix.
And what does that show?
00:44:41.000 --> 00:44:47.000
It shows that the inverse
matrix, the e to the A,
00:44:47.000 --> 00:44:53.000
is e to the negative A.
That is a very useful fact.
00:44:53.000 --> 00:44:59.000
This is the main survivor of
the exponential law.
00:45:00.000 --> 00:45:06.000
In general it is false,
but this standard corollary to
00:45:05.000 --> 00:45:11.000
the exponential law is true,
is equal to e to the minus A,
00:45:10.000 --> 00:45:16.000
just what you would
dream and hope would be true.
00:45:16.000 --> 00:45:22.000
Okay.
I have exactly two and a half
00:45:19.000 --> 00:45:25.000
minutes left in which to do the
impossible.
00:45:23.000 --> 00:45:29.000
All right.
The question is,
00:45:25.000 --> 00:45:31.000
how do you calculate e to the
At?
00:45:31.000 --> 00:45:37.000
You could use series,
but it rarely works.
00:45:34.000 --> 00:45:40.000
It is too hard.
There are a few examples,
00:45:38.000 --> 00:45:44.000
and you will have some more for
homework, but in general it is
00:45:43.000 --> 00:45:49.000
too hard because it is too hard
to calculate the powers of a
00:45:49.000 --> 00:45:55.000
general matrix A.
There is another method,
00:45:52.000 --> 00:45:58.000
which is useful only for
matrices which are symmetric,
00:45:57.000 --> 00:46:03.000
but like that --
Well, it is more than
00:46:01.000 --> 00:46:07.000
symmetric.
These two have to be the same.
00:46:04.000 --> 00:46:10.000
But you can handle those,
as you will see from the
00:46:07.000 --> 00:46:13.000
homework problems,
by breaking it up this way and
00:46:11.000 --> 00:46:17.000
using the exponential law.
This would be zero,
00:46:14.000 --> 00:46:20.000
b, b, zero.
00:46:16.000 --> 00:46:22.000
See, these two matrices commute
with each other and,
00:46:19.000 --> 00:46:25.000
therefore, I could use the
exponential law.
00:46:22.000 --> 00:46:28.000
This leaves all other cases.
And here is the way to handle
00:46:26.000 --> 00:46:32.000
all other cases.
All other cases.
00:46:30.000 --> 00:46:36.000
In other words,
if you cannot calculate the
00:46:33.000 --> 00:46:39.000
series, this trick doesn't work,
I have done as follows.
00:46:38.000 --> 00:46:44.000
You start with an arbitrary
fundamental matrix,
00:46:41.000 --> 00:46:47.000
not the exponential matrix.
You multiply it by its value at
00:46:46.000 --> 00:46:52.000
zero, that is a constant matrix,
and you take the inverse of
00:46:51.000 --> 00:46:57.000
that constant matrix.
It will have one because,
00:46:55.000 --> 00:47:01.000
remember, the fundamental
matrix never has the determinant
00:47:00.000 --> 00:47:06.000
zero.
So you can always take its
00:47:04.000 --> 00:47:10.000
inverse-ready value of t.
Now, what property does this
00:47:09.000 --> 00:47:15.000
have?
It is a fundamental matrix.
00:47:12.000 --> 00:47:18.000
How do I know that?
Well, because I found all
00:47:16.000 --> 00:47:22.000
fundamental matrices for you.
Take any one,
00:47:21.000 --> 00:47:27.000
multiply it by a square matrix
on the right-hand side,
00:47:26.000 --> 00:47:32.000
and you get still a fundamental
matrix.
00:47:29.000 --> 00:47:35.000
And what is its value at zero?
Well, it is x of zero times x
00:47:37.000 --> 00:47:43.000
of zero inverse.
Its value at zero is the
00:47:42.000 --> 00:47:48.000
identity.
Now, e to the At has
00:47:48.000 --> 00:47:54.000
these same two properties.
00:47:56.000 --> 00:48:02.000
Namely, it is a fundamental
matrix and its value at zero is
00:48:01.000 --> 00:48:07.000
the identity. Conclusion,
this is e to the At.
00:48:05.000 --> 00:48:11.000
And that is the garden variety
00:48:08.000 --> 00:48:14.000
method of calculating the
exponential matrix,
00:48:11.000 --> 00:48:17.000
if you want to give it
explicitly.
00:48:13.000 --> 00:48:19.000
Start with any fundamental
matrix calculated,
00:48:16.000 --> 00:48:22.000
you should forgive the
expression using eigenvalues and
00:48:20.000 --> 00:48:26.000
eigenvectors and putting the
solutions into the columns.
00:48:24.000 --> 00:48:30.000
Evaluate it at zero,
take its inverse and multiply
00:48:28.000 --> 00:48:34.000
the two.
And what you end up with has to
00:48:32.000 --> 00:48:38.000
be the same as the thing
calculated with that infinite
00:48:36.000 --> 00:48:42.000
series.
Okay.
You will get lots of practice
for homework and tomorrow.