WEBVTT
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We're going to need a few facts about fundamental matrices and I'm worried that over the weekend
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this spring activities weekend you might have forgotten them.
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So I'll just spend two or three minutes reviewing the most essential things that we're going
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to need later in the period.
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So what we're talking about is I'll try to color code things so you'll know what they
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are.
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First of all, we're the basic problem is to solve a system of equations and I'm going
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to make that a 2 by 2 system, although practically everything I say today will also work for
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N by N systems.
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Your book tries to do it N by N as usual, but I think it's easier to learn 2 by 2 first
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and generalize rather than to way through the complications of N by N systems.
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So the problem is to solve it and the method I used last time was to describe something
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called a fundamental matrix.
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A fundamental matrix for the system of for A, whichever you want.
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Remember what that was.
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That was something whose it was a 2 by 2 matrix of functions of t and whose columns were
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two independent solutions.
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X1, X2, these were two independent solutions.
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In other words, neither was a constant multiple of the other.
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Now I spent a fair amount of time showing you how to the essential, the two essential
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properties that a fundamental matrix had.
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So we're going to need those today.
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So let me remind you the basic properties, the basic properties of X and the properties
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by which you could recognize one if you were given one.
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First of all, the easy one that it's determinant shall not be zero is not zero for any t, for
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any value of the variable.
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That simply expresses the fact that its two columns are independent, linearly independent,
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not a multiple of each other.
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The other one was more bizarre, so I tried to call a little more attention to it.
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It was that the matrix satisfies a differential equation of its own, which looks almost the
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same except it's a matrix differential equation.
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It's not column vectors which are solutions, but matrices as a whole which are solutions.
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In other words, X, if you take that matrix and differentiate every entry, what you get
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is the same as A multiplied by that matrix you started with.
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This remember, express the fact it was just really formal when you analyze what it was,
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but it expresses the fact that the columns, it says that the columns are solve the system.
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So the first thing says the columns are independent and the second say that the columns, each
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column separately, is a solution to the system.
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This as far as more or less, then I went in another direction and we talked about variation
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of parameters.
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I'm not going to come back to variation of parameters today.
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We're going in a different tack.
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The tack we're going on is I want to first talk a little more about the fundamental matrix
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and then as I said, we'll talk about an entirely different method of solving the system, one
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which makes no mention of eigenvalues or eigenvectors if you can believe that.
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But first, the one confusing thing about the fundamental matrix is that it is not unique.
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I've been carefully trying to avoid talking about the fundamental matrix because there
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is no V fundamental matrix.
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There is only V fundamental matrix.
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Why is that?
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Well, because these two columns can be any two independent solutions and they're an
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infinity of ways of picking independent solutions.
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That means they're an infinity of possible fundamental matrices.
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Well, that's disgusting.
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But can we repair it a little bit?
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I mean, maybe they're all derivable from each other in some simple way and that's, of
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course, what is true.
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Now, as a prior to you to doing that, I'd like to show you what I should have showed
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you on Friday, but again, I ran out of time.
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How do you write the general solution to the system?
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The persistent I'm talking about is that pink system.
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Well, of course, the standard naive way of doing it is, it's x equals the general solution
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is an arbitrary constant times that first solution you found plus c2 times the second, another
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arbitrary constant times the second solution you found.
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Okay.
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Now, how would you abbreviate that using the fundamental matrix?
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Well, I did something very similar to this on Friday except these were called V's.
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It was part of the variation of parameters, but I promised not to use those words today.
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So I just said nothing.
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Okay.
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What is the answer?
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So it is x equals, how do I write this using the fundamental matrix?
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x1, x2.
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Simple.
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It's capital X times the vector whose column, the vector whose entry column vector whose
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entries are c1 and c2.
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In other words, it is x1, x2 times the column vector c1, c2, isn't it?
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Yeah, because if you multiply this, think of, think top row, top row, top row, c1, top
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row plus top row times c2, that exactly gives you the top row here and the same way the
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bottom row here times this vector gives you the bottom row of that.
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So it's just another way of writing that, but it looks very efficient.
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Now, sometimes efficiency isn't a good thing.
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You have to watch out for it, but here it's good.
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So this is the general solution.
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Written out using a fundamental matrix and you can't use less symbols than that.
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There's just no way.
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Well, but that gives us our answer to what do all fundamental matrices look like?
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Well, there are two columns, our solutions.
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So the answer is they look like x.
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Now, the first column is an arbitrary solution.
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How do I write an arbitrary solution?
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There's the general solution.
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I make it a particular one by giving a particular value to that column vector of arbitrary
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constants, like two, three, or minus one pi or something like that.
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So if the first guy is a solution and I've just shown you, I can write such a solution
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like x, c, 1 with a column vector, 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'll write another one.
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And now all I have to do is, of course, that's supposed to be a dependent.
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We'll worry about that in just a moment.
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All I have to do is make this look better.
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Now, I showed you last time by the laws of matrix multiplication.
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If the first column is x, c, 1, and the second column is x, c, 2, using matrix multiplication,
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that's the same as writing it this way.
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This square matrix times the matrix whose entries are the first column vector and the
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second column vector.
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Now, I'm going to call this c.
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It's a square matrix of constants, it's a 2 by 2 matrix of constants.
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And so the final way of writing it is x times, it's just what corresponds to that, but x times
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c.
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And so this is the most general, so x is a given fundamental matrix, this one, that one.
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So the most general fundamental matrix is then the one you started with and multiply it
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by an arbitrary square matrix of constants, except you want to be sure that the determinant
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is not zero.
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Well, the determinant of this guy won't be zero, so what you have to do is make sure
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that the determinant of c isn't zero either.
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So in other words, the fundamental matrix isn't unique, but once you found one, all the
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other ones are found by multiplying it on the right by an arbitrary square matrix of
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constants, which is non-singular, has determinant not zero in other words.
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Well, that was all Friday.
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That's Friday leaking over into Monday, and now we begin the true Monday.
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So here's the problem.
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Once again, we've got to start with, we've got to 2 by 2 system or n by n if you want to
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be super general.
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There's a system.
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What have we got so far by way of solving it?
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If somebody, you know, your kid brother or sister, when you go home, says, okay, you know,
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precocious kid, says, okay, so tell me how to solve this thing.
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I think all the only thing you'll be able to say, well, you do this, you take the matrix
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and then you calculate something called eigenvalues and eigenvectors.
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You know what those are?
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Yeah, I didn't think you did.
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Well, I'm smart, I am.
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And you then explain what the eigenvalues and eigenvectors are.
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And then you show how to, of those make up special solutions and then you take a combination
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of that.
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In other words, it's an algorithm.
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It's something you do a process, a method.
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And when it's all done, you have the general solution.
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Now that's fine for calculating particular problems, you know, with definite numbers,
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a definite model, with definite numbers in it where you want a definite answer.
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And of course, a lot of your work in engineering and science classes is that kind of
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work.
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But the further you get on, well, when you start reading books, for example, or Doc would
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bid start reading papers in which, you know, people are telling you, you know, they're
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doing engineering or they're 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, what I, the problem is to fill in the blank in the following.
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The writing of paper, and you just set up some elaborate model in A is a matrix derived
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from that model in some way, you know, represents bacteria doing something or bank accounts
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doing something.
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I don't know.
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And you say, as is well known, the solution is, of course, you've only got letters here.
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No numbers.
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This is a general paper.
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The solution is given by the formula.
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The actual is, we don't have a formula.
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Well, we have as a benefit.
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Now people don't like that.
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So what I'm going to produce for you this period is a formula.
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And that formula does not require the calculation of any eigenvalues, eigenvectors.
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Does it require any of that?
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It's, therefore, a very popular way to fill in the finish that sentence.
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Now the question is, where is that formula going to come from?
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Well, we are, for the moment, clueless.
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If you're clueless, the place to look always is, do I know anything about this sort of
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thing?
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Is it some special case I can solve it, of this problem I can solve, or that I have solved
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in the past?
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And the answer to that is yes.
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You have it solved it for a two by two matrix, but you have solved it for a one by one matrix.
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A one by one matrix also goes by the name of a constant.
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It's just a thing.
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You know, it's a number.
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It's just pretty brackets around, it doesn't conceal that a fact that it's just a number.
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So let's look at what's the solution for a one by one matrix, one by one case.
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If we're looking for a general solution for the n by n case, it must work for the one
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by one case also.
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That's a good reason for a starting.
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That looks like x, doesn't it?
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Since like one by one case.
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Well, in that case, I'm trying to solve the system.
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The system consists of a single equation.
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That's the way the system looks.
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How do you solve that?
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Well, you were born knowing how to solve that, or you anyway certainly didn't learn it
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in this course.
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The solution is, well, you separate variables, blah, blah, blah, blah, blah, blah, blah.
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The solution is x equals, the basic solution is e to the at, and you multiply that by an
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arbitrary constant.
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Now if that's a formula, it's a formula for the solution.
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It uses the parameter in the equation, I didn't have to know a special number, I didn't
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have to put a particular number here to use that.
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Well, the answer is that the same idea, whatever the answer I give here, it's got to work
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in this case too, so I'll just, 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'm going to give you a slightly different argument, which has the advantage of generalizing
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to the n by n case, and the argument goes as follows for that.
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It uses the definition of the exponential function, not as the inverse to the logarithm, which
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is where the fancy calculus books get it from, nor as the naive high school method, you
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know, it's e squared means you multiply e by itself, that e cubed means you do it three
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times, and so on, and either the one half means you do it a half a time or something.
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So the naive definition of the exponential function, instead I'll use the definition
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of the exponential function that comes from an infinite series, so leaving out the arbitrary
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constant, which we don't have to bother with.
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E to the AT is the series 1 plus AT plus a squared, t squared over 2 factorial.
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I'll put out one more term, and let's call it quits there.
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If I take this, and then the argument goes, let's just differentiate it.
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In other words, what's the derivative of e to the AT with respect to t?
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Well, just differentiating term by term, it's 0 plus the first term is a, the next term
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is a squared times t, this differentiates to t squared over 2 factorial, and the answer
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is that this is equal to a times, if you factor out the a, what's left is 1 plus a t plus
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a squared t squared over 2 factorial, in other words, it's simply a, e to the AT.
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In other words, by differentiating the series, using the series definition of the exponential,
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and by differentiating a term by term, I can immediately see that it satisfies this differential
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equation.
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What about the arbitrary constant?
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Well, if you like, you can include it here, but it's easier to observe that by linearity,
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the speed of the AT solves the equation, so does the constant times it, because it's
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linear.
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Now, that's the idea that I'm going to use to solve the system in general.
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What are we going to say?
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Well, what could we say?
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The solution to, well, it's better, let's get two solutions at once by writing a fundamental
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matrix.
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So, a fundamental matrix, a fundamental matrix, I don't play with the one, a fundamental matrix
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for the system x prime equals a x.
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So that's what we're trying to solve, and we're going to get two solutions by getting
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a fundamental matrix for it.
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The answer is, e to the AT.
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Isn't that what it should be?
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00:19:08.400 --> 00:19:12.680
That old little guy had a little a, now he's got a matrix, okay, just put the matrix up
233
00:19:12.680 --> 00:19:13.680
there.
234
00:19:13.680 --> 00:19:18.520
Now, what on earth is, first person who's the most thought of this, it happened about
235
00:19:18.520 --> 00:19:22.280
100 years ago.
236
00:19:22.280 --> 00:19:25.080
What meaning should be given to e to a matrix power?
237
00:19:25.080 --> 00:19:28.920
Well, clearly, the two naive definitions won't work.
238
00:19:28.920 --> 00:19:34.080
The only possible meaning you could try for is using the infinite series, but that does
239
00:19:34.080 --> 00:19:37.400
work.
240
00:19:37.400 --> 00:19:40.920
So this is a definition I'm giving you, the exponential matrix.
241
00:19:40.920 --> 00:19:42.080
It is.
242
00:19:42.080 --> 00:19:47.080
Now, notice the a is a two by two matrix, multiplying it by t.
243
00:19:47.080 --> 00:19:50.320
What I have up here is basically a two by two matrix.
244
00:19:50.320 --> 00:19:54.240
It's entries involved t, but it's a two by two matrix.
245
00:19:54.240 --> 00:20:00.120
Okay, it's going to be, leave out, so we're trying to get the analog of that formula over
246
00:20:00.120 --> 00:20:01.120
there.
247
00:20:01.120 --> 00:20:05.160
Well, it's going to be, leave the first term out just for the moment.
248
00:20:05.160 --> 00:20:09.440
The next term is going to surely be a times t.
249
00:20:09.440 --> 00:20:11.960
This is a two by two matrix, right?
250
00:20:11.960 --> 00:20:12.960
Which of the next term be?
251
00:20:12.960 --> 00:20:17.120
Well, a squared times t squared over two factorial.
252
00:20:17.120 --> 00:20:19.120
What kind of a guy is that?
253
00:20:19.120 --> 00:20:24.520
Well, if a is a two by two matrix, so is a squared.
254
00:20:24.520 --> 00:20:25.520
How about this?
255
00:20:25.520 --> 00:20:32.520
This is just a scalar which multiplies every entry of a squared, and therefore this is still
256
00:20:32.520 --> 00:20:34.680
a two by two matrix.
257
00:20:34.680 --> 00:20:36.640
That's a two by two matrix.
258
00:20:36.640 --> 00:20:38.360
This is a two by two matrix.
259
00:20:38.360 --> 00:20:42.960
No matter how many times you multiply a by itself, it stays a two by two matrix.
260
00:20:42.960 --> 00:20:48.000
It gets more and more complicated looking, but it's always a two by two matrix.
261
00:20:48.000 --> 00:20:53.480
And now I'm multiplying every entry of that by t cube, the scalar t cube over three factorial.
262
00:20:53.480 --> 00:21:00.800
And continuing on in that way, what I get, therefore, is a sum of two by two matrices.
263
00:21:00.800 --> 00:21:04.120
Well, you can add two by two matrices to each other.
264
00:21:04.120 --> 00:21:08.320
We've never made an infinite series of them, but in fact, we haven't done it.
265
00:21:08.320 --> 00:21:10.320
But others have.
266
00:21:10.320 --> 00:21:12.120
And this is what they wrote.
267
00:21:12.120 --> 00:21:15.040
The only question is, what should we put at the beginning?
268
00:21:15.040 --> 00:21:17.760
Over there, I have one.
269
00:21:17.760 --> 00:21:23.400
The number one, but I, of course, cannot add the number one to two by two matrices.
270
00:21:23.400 --> 00:21:29.680
What goes here must be a two by two matrix, which is the closest thing to one I can think of.
271
00:21:29.680 --> 00:21:31.400
What should it be?
272
00:21:31.400 --> 00:21:34.400
The i, two by two i.
273
00:21:34.400 --> 00:21:36.520
The two by two identity matrix.
274
00:21:36.520 --> 00:21:39.440
It looks like the natural candidate for what to put there.
275
00:21:39.440 --> 00:21:44.560
And in fact, it's the right thing to put there.
276
00:21:44.560 --> 00:21:48.480
OK, now I have a conjecture.
277
00:21:48.480 --> 00:21:55.240
You know, purely formally, changing all the, with a key stroke of the computer, all the
278
00:21:55.240 --> 00:21:57.880
little a's have been changed to capital a's.
279
00:21:57.880 --> 00:22:01.240
And now all I have to do is wonder if it's this going to work.
280
00:22:01.240 --> 00:22:06.080
Well, what's the basic thing I have to check to see that it's a fundamental matrix?
281
00:22:06.080 --> 00:22:14.760
So the question is, is this, I wrote it down all right, but is this a fundamental matrix
282
00:22:14.760 --> 00:22:16.640
for the system?
283
00:22:16.640 --> 00:22:21.640
Well, I have a way of recognizing a fundamental matrix when I see one.
284
00:22:21.640 --> 00:22:27.880
The critical thing is that it should satisfy this matrix differential equation.
285
00:22:27.880 --> 00:22:29.920
So that's what I should verify.
286
00:22:29.920 --> 00:22:33.880
Does this guy that I've written down satisfy that equation?
287
00:22:33.880 --> 00:22:47.320
And the answer is it satisfies, so number two is it satisfies x prime equals a x.
288
00:22:47.320 --> 00:22:53.560
In other words, plugging in x equals this e to the a t, who's definition I just gave
289
00:22:53.560 --> 00:22:54.560
you.
290
00:22:54.560 --> 00:22:58.840
If I substitute that in, does it satisfy that matrix differential equation?
291
00:22:58.840 --> 00:22:59.840
The answer is yes.
292
00:22:59.840 --> 00:23:03.200
I'm not going to calculate it out because the calculation is absolutely identical to what
293
00:23:03.200 --> 00:23:05.440
I did there.
294
00:23:05.440 --> 00:23:09.880
The only difference is when I differentiated a term by term, how do you differentiate something
295
00:23:09.880 --> 00:23:10.880
like this?
296
00:23:10.880 --> 00:23:17.600
Well, you differentiate every term in it, but if you work it out, it's simply, this is
297
00:23:17.600 --> 00:23:22.360
a constant matrix, every term of which is multiplied by t squared over two factorial.
298
00:23:22.360 --> 00:23:27.080
Well, if you differentiate every term, every entry of that constant of that matrix, what
299
00:23:27.080 --> 00:23:31.920
you're going to get is a squared times just the derivative of that part, which is simply
300
00:23:31.920 --> 00:23:32.920
t.
301
00:23:32.920 --> 00:23:38.880
In other words, the formal calculation looks absolutely identical to that.
302
00:23:38.880 --> 00:23:44.080
So the answer to this is yes.
303
00:23:44.080 --> 00:23:49.200
By the same calculation as before.
304
00:23:49.200 --> 00:23:54.800
As for the one by one case.
305
00:23:54.800 --> 00:23:59.160
I know the only other thing to check is that the determinant is not zero.
306
00:23:59.160 --> 00:24:02.720
In fact, the determinant is not zero at one point.
307
00:24:02.720 --> 00:24:04.000
That's the one you have to check.
308
00:24:04.000 --> 00:24:06.440
What's x of zero?
309
00:24:06.440 --> 00:24:13.080
What's the determinant at the value of the determinant of x is e to the at.
310
00:24:13.080 --> 00:24:17.040
What's the value of this thing at zero?
311
00:24:17.040 --> 00:24:18.040
Here's my function.
312
00:24:18.040 --> 00:24:21.200
If I plug in t equals zero, what's it equal to?
313
00:24:21.200 --> 00:24:24.600
I, what's the determinant of I?
314
00:24:24.600 --> 00:24:39.080
It is certainly not zero.
315
00:24:39.080 --> 00:24:43.840
So by writing down this infinite series, I've got my two solutions.
316
00:24:43.840 --> 00:24:48.680
Its columns give me two solutions to the original system.
317
00:24:48.680 --> 00:24:52.160
There were no eigenvalues, no eigenvectors.
318
00:24:52.160 --> 00:24:54.800
I have a formula for the answer, what's the formula?
319
00:24:54.800 --> 00:24:55.960
It's e to the at.
320
00:24:55.960 --> 00:24:59.640
And of course, anybody reading the paper is supposed to know what e to the at is.
321
00:24:59.640 --> 00:25:03.400
It's that.
322
00:25:03.400 --> 00:25:04.560
This is just marvelous.
323
00:25:04.560 --> 00:25:08.400
There must be a fly in the ointment somewhere.
324
00:25:08.400 --> 00:25:10.520
Only a t they'll fly.
325
00:25:10.520 --> 00:25:16.840
The t they'll fly is almost impossible to calculate that series for all reasonable times.
326
00:25:16.840 --> 00:25:18.680
However, once in a while it is.
327
00:25:18.680 --> 00:25:23.880
So let me give you an example where it is possible to calculate the series and we get a nice
328
00:25:23.880 --> 00:25:25.640
answer.
329
00:25:25.640 --> 00:25:34.080
So let's work out an example.
330
00:25:34.080 --> 00:25:46.360
By the way, I should, you know, nowadays we're not back 50 years.
331
00:25:46.360 --> 00:25:51.480
And the exponential matrix has the same status on say a MATLAB or MATLAB or MATLAB or
332
00:25:51.480 --> 00:25:52.480
MATLAB.
333
00:25:52.480 --> 00:25:58.080
As the ordinary exponential function does, it's just a command you type in.
334
00:25:58.080 --> 00:26:05.480
You type in your matrix and you now say X, X, E, X, P of that matrix and comes the answer
335
00:26:05.480 --> 00:26:07.440
to as many decimal places as you want.
336
00:26:07.440 --> 00:26:13.760
It will be a square matrix with entries carefully written out.
337
00:26:13.760 --> 00:26:19.240
So in that sense, you know, the fact that we can't calculate it shouldn't bother us,
338
00:26:19.240 --> 00:26:22.000
you know, there are machines to do the calculations.
339
00:26:22.000 --> 00:26:24.960
But we're interested in it as a theoretical tool.
340
00:26:24.960 --> 00:26:30.600
But in order to get any feeling for this at all, we certainly have to do a few calculations.
341
00:26:30.600 --> 00:26:32.280
So let's do an easy one.
342
00:26:32.280 --> 00:26:38.920
Let's consider the system X prime equals Y, Y prime equals X.
343
00:26:38.920 --> 00:26:43.320
This is very easily done by elimination, but that's forbidden.
344
00:26:43.320 --> 00:26:46.120
So first of all, we write it as a matrix.
345
00:26:46.120 --> 00:26:53.520
So X prime, it's the system X prime equals 0, 1, 1, 0, X.
346
00:26:53.520 --> 00:27:04.680
So here's my A. And so E to the AT is going to be, so A is 0, 1, 1, 0.
347
00:27:04.680 --> 00:27:09.600
So what we want to calculate is we're going to get both solutions at once by calculating
348
00:27:09.600 --> 00:27:12.400
a one-shell swoop E to the AT.
349
00:27:12.400 --> 00:27:16.720
Okay, E to the AT equals.
350
00:27:16.720 --> 00:27:20.160
Okay, I'm going to actually write out these guys.
351
00:27:20.160 --> 00:27:25.840
Well, obviously the hard part of the part which is normally going to prevent us from calculating
352
00:27:25.840 --> 00:27:31.080
this series explicitly by hand anyway, as I say, the computer can always do it.
353
00:27:31.080 --> 00:27:36.400
Is the value, how do we raise a matrix to a high power?
354
00:27:36.400 --> 00:27:38.800
You just keep multiplying and multiplying and multiplying.
355
00:27:38.800 --> 00:27:42.840
That looks like a rather forbidding and unpromising activity.
356
00:27:42.840 --> 00:27:44.520
Well, here it's easy.
357
00:27:44.520 --> 00:27:45.600
Let's see what happens.
358
00:27:45.600 --> 00:27:48.800
What is A, if that's A, what's A squared?
359
00:27:48.800 --> 00:27:51.560
I'm going to have to calculate that as part of the series.
360
00:27:51.560 --> 00:28:08.360
So that's going to be 0, 1, 1, 0, times 0, 1, 1, 0, which is, so 1, 0, 0, 0, 0, 1, plus
361
00:28:08.360 --> 00:28:12.400
0, we are saved.
362
00:28:12.400 --> 00:28:15.880
It is the identity.
363
00:28:15.880 --> 00:28:19.480
Now for this point on, we don't have to do any more calculations, but I'll do the many
364
00:28:19.480 --> 00:28:24.440
way just to, what's A cubed?
365
00:28:24.440 --> 00:28:26.080
Don't start from scratch again.
366
00:28:26.080 --> 00:28:29.760
God, I want to know, no, no, no, no.
367
00:28:29.760 --> 00:28:36.600
A cubed is A squared times A, and A squared is in real life the identity.
368
00:28:36.600 --> 00:28:40.160
Of course you would do all this in your head, but I'm being a good boy writing in a life.
369
00:28:40.160 --> 00:28:47.080
So this is I, the identity, times A, which is A. I'll do one more.
370
00:28:47.080 --> 00:28:48.760
What's A to the fourth?
371
00:28:48.760 --> 00:28:54.640
Now you would be tempted to say, A to the fourth is A squared, which is I times I, which
372
00:28:54.640 --> 00:28:57.120
is I, but that would be wrong.
373
00:28:57.120 --> 00:29:07.680
A to the fourth is A cubed times A, which is A cubed, I've just calculated is A times
374
00:29:07.680 --> 00:29:12.320
A. A cubed, I've just calculated, it was A, right?
375
00:29:12.320 --> 00:29:18.520
And now that is A squared, which is the identity.
376
00:29:18.520 --> 00:29:23.960
It's clear this is, by this argument, it's going to continue in the same way each time
377
00:29:23.960 --> 00:29:29.320
you add an A on the right hand side, you're going to keep alternating between the identity
378
00:29:29.320 --> 00:29:32.480
A, the next one will be the identity, the next will be A.
379
00:29:32.480 --> 00:29:41.280
So the end result is that the first term of the series is simply the identity, the next
380
00:29:41.280 --> 00:29:47.880
term of the series is A, but it's multiplied by T. So that is zero.
381
00:29:47.880 --> 00:29:53.760
So I put it, I'll keep the T at the outside.
382
00:29:53.760 --> 00:29:58.360
Remember when you multiply a matrix by a scalar, that means multiply every entry by that
383
00:29:58.360 --> 00:29:59.360
scalar.
384
00:29:59.360 --> 00:30:03.400
So this is the matrix zero, T, T, zero.
385
00:30:03.400 --> 00:30:05.240
I'll do a couple more terms.
386
00:30:05.240 --> 00:30:09.560
The next term would be, well, A squared, we just calculated is the identity.
387
00:30:09.560 --> 00:30:12.160
So that looks like this.
388
00:30:12.160 --> 00:30:16.000
Except now I multiply every term by T squared over 2 factorial.
389
00:30:16.000 --> 00:30:20.280
Oh, all right, let's go for broke.
390
00:30:20.280 --> 00:30:25.000
The next one will be this, times T cubed over 3 factorial.
391
00:30:25.000 --> 00:30:26.480
Fortunately, I've run out of room.
392
00:30:32.360 --> 00:30:33.680
Okay, let's calculate then.
393
00:30:37.600 --> 00:30:39.800
Maybe I should do it.
394
00:30:39.800 --> 00:30:42.480
Well, no, we're going to need that.
395
00:30:42.480 --> 00:30:54.240
Okay, let's do it here.
396
00:30:54.240 --> 00:30:56.400
So what's the final answer for E to the AT?
397
00:30:56.400 --> 00:31:00.840
It is, I have an infinite series of two by two matrices.
398
00:31:00.840 --> 00:31:04.200
Let's look at the term in the upper left hand corner.
399
00:31:04.200 --> 00:31:15.360
It's 1 plus 0 times T plus 1 times T squared over 2 factorial plus 0 times T, it's going
400
00:31:15.360 --> 00:31:21.760
to another words to be 1 plus T squared over 2 factorial plus the next term might not
401
00:31:21.760 --> 00:31:24.600
on the board, but I think you can see it's this.
402
00:31:24.600 --> 00:31:26.960
And it continues on in the same way.
403
00:31:26.960 --> 00:31:28.920
How about the lower left term?
404
00:31:28.920 --> 00:31:38.120
Well, that's 0 plus T plus 0 plus T cubed over 3 factorial and so on.
405
00:31:38.120 --> 00:31:45.240
So it's T plus T cubed over 3 factorial plus T of the 5th over 5 factorial.
406
00:31:45.240 --> 00:31:49.720
The other terms in the other two corners are just the same as these.
407
00:31:49.720 --> 00:31:56.960
This one, for example, is 0 plus T plus 0 plus T cubed over 3 factorial.
408
00:31:56.960 --> 00:32:02.040
And the lower one is 1 plus 0 plus T squared and so on.
409
00:32:02.040 --> 00:32:09.840
So it's the same as 1 plus T squared over 2 factorial and so on.
410
00:32:09.840 --> 00:32:15.000
And up here we have T plus T cubed over 3 factorial and so on.
411
00:32:18.200 --> 00:32:22.240
Well, that matrix doesn't look very square, but it is.
412
00:32:22.240 --> 00:32:26.760
It looks, it's infinitely log physically, but it isn't.
413
00:32:26.760 --> 00:32:31.800
It's got one term here, one term here, one term here, one term there.
414
00:32:31.800 --> 00:32:35.800
Now, all we have to do is make those terms look a little better.
415
00:32:35.800 --> 00:32:40.520
For here, I have to rely on the culture which you may or may not possess.
416
00:32:44.720 --> 00:32:49.840
You would know what these series were if only they alternated their signs.
417
00:32:49.840 --> 00:32:56.000
If this were a negative, negative, negative, then the top would be cosine T.
418
00:32:56.000 --> 00:32:59.920
And this would be sine T, but they don't.
419
00:33:02.120 --> 00:33:03.680
So they are the next best thing.
420
00:33:03.680 --> 00:33:08.200
They are hyperbolic.
421
00:33:08.200 --> 00:33:10.360
Right?
422
00:33:10.360 --> 00:33:14.640
The top is not cosine T, but cos T.
423
00:33:14.640 --> 00:33:22.320
The bottom is 6T, and how do we know this because you remember?
424
00:33:22.320 --> 00:33:25.120
And what if I don't remember?
425
00:33:25.120 --> 00:33:27.800
Well, you know now.
426
00:33:27.800 --> 00:33:29.120
That's why you come to class.
427
00:33:35.440 --> 00:33:40.200
Well, for those of you who don't, and in fact, remember the nice way to,
428
00:33:40.200 --> 00:33:43.600
so this is E to the T plus, E to the negative T.
429
00:33:43.600 --> 00:33:47.760
It should be over 2, but I don't have room to put in the 2.
430
00:33:47.760 --> 00:33:49.880
This doesn't mean I will omitted.
431
00:33:49.880 --> 00:34:00.640
It just means I will put it in at the end by multiplying every entry of this matrix by 1.5.
432
00:34:05.440 --> 00:34:11.440
So if you're forgotten what cos T is, it's E to the T plus E to the negative T divided by 2,
433
00:34:11.440 --> 00:34:14.200
and the same similar thing for sine T.
434
00:34:17.000 --> 00:34:22.760
So there is your first explicit exponential matrix calculated according to the definition.
435
00:34:22.760 --> 00:34:32.160
And what we have found is the solution to the system X prime equals Y, Y prime equals X.
436
00:34:32.160 --> 00:34:41.400
A fundamental matrix, in other words, our cos T and sine T satisfy both solutions of that system.
437
00:34:41.400 --> 00:34:47.360
Now, there's one thing people love the exponential matrix in particular for,
438
00:34:47.360 --> 00:34:51.840
and that is the E's with which it solves the initial value problem.
439
00:34:51.840 --> 00:34:58.240
It's exactly what happens in studying the single system, the single equation, X prime equals AX.
440
00:34:58.240 --> 00:35:02.560
But let's do it in general.
441
00:35:02.560 --> 00:35:04.960
Let's do it in general.
442
00:35:04.960 --> 00:35:06.640
What's the initial value problem?
443
00:35:06.640 --> 00:35:14.200
Well, the initial value problem is we start with our old system, but now I want to put it plug in initial conditions.
444
00:35:14.200 --> 00:35:19.640
I want the particular solution which satisfies the initial condition.
445
00:35:19.640 --> 00:35:25.560
Let's make it a zero to avoid complications, to avoid a lot of notation.
446
00:35:25.560 --> 00:35:28.560
This is to be some starting value.
447
00:35:28.560 --> 00:35:31.480
So this is a constant, certain constant vector.
448
00:35:31.480 --> 00:35:35.120
It's to be the value of the solution at zero.
449
00:35:35.120 --> 00:35:43.640
And the problem is find what X of T is.
450
00:35:43.640 --> 00:35:48.320
Well, if you're using the exponential matrix, it's a joke.
451
00:35:48.320 --> 00:35:50.600
It's a joke.
452
00:35:50.600 --> 00:35:53.600
So I derive it or just do it.
453
00:35:53.600 --> 00:36:02.800
All right, the general solution, let's derive it and then I'll put up the final formula in a box so that you'll know it's important.
454
00:36:02.800 --> 00:36:04.200
What's the general solution?
455
00:36:04.200 --> 00:36:07.720
Well, I did that for you at the beginning of the period.
456
00:36:07.720 --> 00:36:15.440
Once you have a fundamental matrix, you get the general solution by multiplying it on the right by an arbitrary constant vector.
457
00:36:15.440 --> 00:36:21.560
So the general solution is going to be X equals E to the AT.
458
00:36:21.560 --> 00:36:24.040
That's my super fundamental matrix.
459
00:36:24.040 --> 00:36:27.400
Found without eigenvalues and eigenvectors.
460
00:36:27.400 --> 00:36:32.480
And this should be multiplied by some unknown constant vector, C.
461
00:36:32.480 --> 00:36:35.680
So the only question is what should the constant vector be?
462
00:36:35.680 --> 00:36:40.960
So we'll plug in to find C, I'll plug in zero.
463
00:36:40.960 --> 00:36:48.440
When T is zero, here I get X of zero, here I get E to the A times zero times C.
464
00:36:48.440 --> 00:36:49.560
Now what's this?
465
00:36:49.560 --> 00:36:52.880
This is the vector of initial conditions.
466
00:36:52.880 --> 00:36:55.840
What's E to the A times zero?
467
00:36:55.840 --> 00:36:58.480
E to the A times zero.
468
00:36:58.480 --> 00:37:00.320
Plug in T equals zero.
469
00:37:00.320 --> 00:37:02.320
What do you get?
470
00:37:02.320 --> 00:37:03.320
I.
471
00:37:03.320 --> 00:37:04.320
I.
472
00:37:08.320 --> 00:37:12.320
Therefore, C is what?
473
00:37:12.320 --> 00:37:14.120
C is X zero.
474
00:37:14.120 --> 00:37:16.360
It's a total joke.
475
00:37:16.360 --> 00:37:20.080
It's a total joke and the solution is,
476
00:37:20.080 --> 00:37:32.640
the initial value problem is X equals E to the AT times X zero.
477
00:37:32.640 --> 00:37:35.360
It's just what it would have been.
478
00:37:35.360 --> 00:37:36.600
It was variable.
479
00:37:36.600 --> 00:37:41.040
The only difference is that here we're allowed to put the C out front.
480
00:37:41.040 --> 00:37:43.720
In other words, if I asked you to put it in the initial condition,
481
00:37:43.720 --> 00:37:49.880
you'd probably write X equals little X zero times E to the AT.
482
00:37:49.880 --> 00:37:52.160
You would be tempted to do the same thing here.
483
00:37:52.160 --> 00:37:56.800
X equals vector X equals vector X zero times E to the AT.
484
00:37:56.800 --> 00:37:59.000
Now you can't do that.
485
00:37:59.000 --> 00:38:04.240
If you try to, you know, MATLAB will hiccup and say, illegal operation.
486
00:38:04.240 --> 00:38:06.080
What's the illegal operation?
487
00:38:06.080 --> 00:38:08.880
Well, X is a column vector.
488
00:38:08.880 --> 00:38:13.240
From the system, it's a column vector.
489
00:38:13.240 --> 00:38:16.520
That means the initial conditions are also a column vector.
490
00:38:16.520 --> 00:38:24.120
You cannot multiply a column vector out front and the square matrix afterwards.
491
00:38:24.120 --> 00:38:26.240
You can't.
492
00:38:26.240 --> 00:38:28.280
If you want to multiply a matrix by a column vector,
493
00:38:28.280 --> 00:38:29.440
it has to come afterwards.
494
00:38:29.440 --> 00:38:31.560
So you can do ZING, ZING.
495
00:38:31.560 --> 00:38:33.240
There is no ZING.
496
00:38:33.240 --> 00:38:34.240
You see, there is no.
497
00:38:34.240 --> 00:38:35.200
You can't put it in front.
498
00:38:35.200 --> 00:38:37.800
It doesn't work.
499
00:38:37.800 --> 00:38:39.120
So it must go behind.
500
00:38:39.120 --> 00:38:42.880
That's the only place you might get stripped up.
501
00:38:42.880 --> 00:38:45.640
And as I say, if you try to type that in using MATLAB,
502
00:38:45.640 --> 00:38:49.360
you will immediately get error messages.
503
00:38:49.360 --> 00:38:50.040
It's illegal.
504
00:38:50.040 --> 00:38:50.880
Can't do that.
505
00:38:50.880 --> 00:38:54.200
So anyway, we've got our solution.
506
00:38:54.200 --> 00:38:57.080
So here's our system.
507
00:38:57.080 --> 00:39:04.160
Let's say it's our initial value problem anyway is in pink and its solution
508
00:39:04.160 --> 00:39:08.360
using the exponential matrix is in green.
509
00:39:08.360 --> 00:39:14.520
Now, the only problem is we still have to talk a little bit more about calculating
510
00:39:14.520 --> 00:39:17.840
the values.
511
00:39:17.840 --> 00:39:25.600
Now, the principle warning with an exponential matrix is that once you've gotten by the simplest
512
00:39:25.600 --> 00:39:29.800
things involving it, the fact that it solves systems, gives you the fundamental matrix
513
00:39:29.800 --> 00:39:34.440
for a system, then you start flexing your muscles and say, oh, well, let's see what else
514
00:39:34.440 --> 00:39:36.440
we can do with this.
515
00:39:36.440 --> 00:39:42.160
For example, the reason exponentials came into being in the first place was because of
516
00:39:42.160 --> 00:39:44.760
the exponential law, right?
517
00:39:44.760 --> 00:39:50.360
I'll kill anybody who sends the e-mail, say, here was the exponential law.
518
00:39:50.360 --> 00:39:55.680
Okay, the exponential law would say that e to the a plus b is equal to e to the a times
519
00:39:55.680 --> 00:39:59.040
e to the b.
520
00:39:59.040 --> 00:40:02.920
The law of exponents, in other words, it's the thing which makes the exponential function
521
00:40:02.920 --> 00:40:04.840
different from all other functions.
522
00:40:04.840 --> 00:40:06.720
It satisfies something like that.
523
00:40:06.720 --> 00:40:09.600
Now, first of all, does this make sense?
524
00:40:09.600 --> 00:40:12.480
That is the symbol's compatible.
525
00:40:12.480 --> 00:40:14.080
Let's see.
526
00:40:14.080 --> 00:40:16.480
This is a 2 by 2 matrix.
527
00:40:16.480 --> 00:40:21.320
This is a 2 by 2 matrix, so it does make sense to multiply them and the answer will be
528
00:40:21.320 --> 00:40:23.280
a 2 by 2 matrix.
529
00:40:23.280 --> 00:40:24.640
How about here?
530
00:40:24.640 --> 00:40:26.480
This is a 2 by 2 matrix.
531
00:40:26.480 --> 00:40:27.480
Add this to it.
532
00:40:27.480 --> 00:40:29.560
It's still a 2 by 2 matrix.
533
00:40:29.560 --> 00:40:33.320
e to a 2 by 2 matrix still comes out of e to a 2 by 2 matrix.
534
00:40:33.320 --> 00:40:36.880
So both sides are legitimate 2 by 2 matrices.
535
00:40:36.880 --> 00:40:39.760
My only question is, are they equal?
536
00:40:39.760 --> 00:40:47.720
The answer is, not in a pig's eye.
537
00:40:47.720 --> 00:40:48.720
How could this be?
538
00:40:48.720 --> 00:40:50.640
Well, I didn't make up these laws.
539
00:40:50.640 --> 00:40:57.360
I just obeyed them.
540
00:40:57.360 --> 00:41:01.920
I wish I had time to do a little calculation to show you that it's not true.
541
00:41:01.920 --> 00:41:06.880
So it is true in a certain special cases.
542
00:41:06.880 --> 00:41:12.040
It's true in the special case.
543
00:41:12.040 --> 00:41:16.240
And this is pretty much if and only if statement.
544
00:41:16.240 --> 00:41:24.200
The only case in which it's true is if a and b are non-arbitrary square matrices,
545
00:41:24.200 --> 00:41:27.640
but commute with each other.
546
00:41:27.640 --> 00:41:32.040
And if you start writing out the series to try to check whether that law is true, you'll
547
00:41:32.040 --> 00:41:33.560
get a bunch of terms here.
548
00:41:33.560 --> 00:41:35.240
A bunch of terms here.
549
00:41:35.240 --> 00:41:40.240
And you'll find that those terms appear wise equal only if you're allowed to let the
550
00:41:40.240 --> 00:41:42.560
matrices commute with each other.
551
00:41:42.560 --> 00:41:49.400
In other words, if you can turn a b plus b a into twice a b, then everything will work
552
00:41:49.400 --> 00:41:50.400
fine.
553
00:41:50.400 --> 00:41:53.120
But if you can't do that, it will not.
554
00:41:53.120 --> 00:41:57.640
Now, when do two square matrices commute with each other?
555
00:41:57.640 --> 00:41:59.760
The answer is almost never.
556
00:41:59.760 --> 00:42:03.120
It's just a lucky accident if they do.
557
00:42:03.120 --> 00:42:10.800
But there are three cases of the lucky accident which you should know.
558
00:42:10.800 --> 00:42:17.000
So the three cases, I feel justified in calling it the three cases.
559
00:42:17.000 --> 00:42:20.480
The three, oh well.
560
00:42:20.480 --> 00:42:25.720
I don't know if it should do that.
561
00:42:25.720 --> 00:42:30.440
So the three most significant are examples are.
562
00:42:30.440 --> 00:42:35.480
Example number one, when a is a constant times the identity matrix.
563
00:42:35.480 --> 00:42:40.480
In other words, when a is a matrix which looks like this, that matrix commutes with every
564
00:42:40.480 --> 00:42:42.760
other square matrix.
565
00:42:42.760 --> 00:42:50.440
If that's a, then this law a b b a is always true and you're allowed to use this.
566
00:42:50.440 --> 00:42:53.880
So that's one case.
567
00:42:53.880 --> 00:42:59.760
Another case when a is more general is when b is equal to negative a.
568
00:42:59.760 --> 00:43:06.560
I think you can see that that's going to work because a times minus a is equal to minus
569
00:43:06.560 --> 00:43:07.560
a times a.
570
00:43:07.560 --> 00:43:13.600
Yeah, they're both equal to a squared except with a negative sign in front.
571
00:43:13.600 --> 00:43:23.960
So the third case is when b is equal to the inverse of a because a a inverse is the same
572
00:43:23.960 --> 00:43:28.120
as a inverse a, they're both the identity.
573
00:43:28.120 --> 00:43:30.760
Of course a must have an inverse.
574
00:43:30.760 --> 00:43:32.480
Okay, let's suppose it does.
575
00:43:32.480 --> 00:43:39.600
Now of them, this is the, I think the most important one because it leads to this law
576
00:43:39.600 --> 00:43:43.440
which is the form in which you shouldn't.
577
00:43:43.440 --> 00:43:48.400
That's forbidden, but there's one case of it which is forbidden, which is not forbidden,
578
00:43:48.400 --> 00:43:49.800
and that's here.
579
00:43:49.800 --> 00:43:51.120
What will it say?
580
00:43:51.120 --> 00:44:00.600
Well, that will say that e to the a minus a is equal to e to the a times e to the negative
581
00:44:00.600 --> 00:44:03.040
a.
582
00:44:03.040 --> 00:44:07.600
So this is true, even though the general law is false.
583
00:44:07.600 --> 00:44:11.200
Because a and negative a commute with each other.
584
00:44:11.200 --> 00:44:12.440
But now what does this say?
585
00:44:12.440 --> 00:44:15.880
What's e to the zero matrix?
586
00:44:15.880 --> 00:44:21.840
In other words, suppose I take the matrix that zero and plug it into the formula for e.
587
00:44:21.840 --> 00:44:23.280
What do you get?
588
00:44:23.280 --> 00:44:26.920
e to the zero times t is i.
589
00:44:26.920 --> 00:44:31.360
It has to be a two by two matrix, so it's going to be anything.
590
00:44:31.360 --> 00:44:32.720
It's the matrix i.
591
00:44:32.720 --> 00:44:43.120
So this side is i, this side is the exponential matrix, and what does that show?
592
00:44:43.120 --> 00:44:52.680
It shows that the inverse matrix to e to the a is e to the negative a.
593
00:44:52.680 --> 00:44:54.160
That's a very useful fact.
594
00:44:54.160 --> 00:45:01.360
That the inverse to, this is the main survivor of the exponential law.
595
00:45:01.360 --> 00:45:07.360
So it's false, but this standard carl area, the exponential law, is true, is equal to
596
00:45:07.360 --> 00:45:09.040
e to the minus a.
597
00:45:09.040 --> 00:45:12.440
Just what you would dream and hope would be true.
598
00:45:12.440 --> 00:45:24.120
Okay, I've got exactly two and a half minutes left, which to do the impossible.
599
00:45:24.120 --> 00:45:36.680
The question is how do you, to calculate e to the a t, you can use series but rarely
600
00:45:36.680 --> 00:45:37.680
works.
601
00:45:37.680 --> 00:45:41.240
It's too hard.
602
00:45:41.240 --> 00:45:43.200
You know, there are a few examples.
603
00:45:43.200 --> 00:45:44.880
You'll have some more for homework.
604
00:45:44.880 --> 00:45:49.120
But in general, it's too hard because it's too hard to calculate the powers of a general
605
00:45:49.120 --> 00:45:51.120
matrix a.
606
00:45:51.120 --> 00:45:58.480
There's another method which depends, is useful only for matrices which are symmetric.
607
00:45:58.480 --> 00:46:02.200
But like that, well, it's more than symmetric.
608
00:46:02.200 --> 00:46:03.640
These two have to be the same.
609
00:46:03.640 --> 00:46:08.400
But you can handle those as you'll see from the homework problems by breaking it up this
610
00:46:08.400 --> 00:46:11.360
way and using the exponential law.
611
00:46:11.360 --> 00:46:14.760
So this would be zero, b, b, zero.
612
00:46:14.760 --> 00:46:21.760
Basically these two matrices compute with each other and therefore I can use the exponential
613
00:46:21.760 --> 00:46:23.760
law.
614
00:46:23.760 --> 00:46:25.880
This leaves all other cases.
615
00:46:25.880 --> 00:46:29.400
And here's the way to handle all other cases.
616
00:46:29.400 --> 00:46:30.400
All other cases.
617
00:46:30.400 --> 00:46:38.720
In other words, if you can't calculate the series and this trick doesn't work, all done
618
00:46:38.720 --> 00:46:44.880
as follows.
619
00:46:44.880 --> 00:46:47.960
Not the exponential matrix.
620
00:46:47.960 --> 00:46:54.040
You multiply it by its value at zero.
621
00:46:54.040 --> 00:46:58.040
That's a constant matrix and you take the inverse of that constant matrix.
622
00:46:58.040 --> 00:47:02.520
It will have one because remember the fundamental matrix never has a determinant zero.
623
00:47:02.520 --> 00:47:06.040
So you can always take its inverse for any value of t.
624
00:47:06.040 --> 00:47:10.360
Now what property does this have?
625
00:47:10.360 --> 00:47:13.560
It's a fundamental matrix.
626
00:47:13.560 --> 00:47:14.880
How do I know that?
627
00:47:14.880 --> 00:47:18.520
Well because I found all fundamental matrices for you.
628
00:47:18.520 --> 00:47:24.040
Take any one, multiply it by a square matrix on the right hand side and you get still a
629
00:47:24.040 --> 00:47:29.120
fundamental matrix.
630
00:47:29.120 --> 00:47:41.520
And once it's value at zero, well it is x of zero times x of zero inverse.
631
00:47:41.520 --> 00:47:47.320
So its value at zero is the identity.
632
00:47:47.320 --> 00:47:57.280
Now e to the a t has the same two properties.
633
00:47:57.280 --> 00:48:04.560
Similarly, it's a fundamental matrix and its value at zero is the identity.
634
00:48:04.560 --> 00:48:09.120
Conclusion, this is e to the a t.
635
00:48:09.120 --> 00:48:14.080
In other words, and that's the garden variety method of calculating the exponential matrix.
636
00:48:14.080 --> 00:48:20.240
If you want to give it explicitly, start with any fundamental matrix calculated.
637
00:48:20.240 --> 00:48:24.920
You should forget the expression using eigenvalues and eigenvectors and putting the solutions
638
00:48:24.920 --> 00:48:27.200
into the columns.
639
00:48:27.200 --> 00:48:32.040
You get a zero, take its inverse, multiply the two and what you end up with has to be
640
00:48:32.040 --> 00:48:36.040
the same as the thing calculated without infinite series.
641
00:48:36.040 --> 00:49:06.000
Okay, you'll get lots of practice for homework and tomorrow.