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5, 4.
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Okay, this is the lecture on positive definite matrices.
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I made a start on those briefly in a previous lecture and one point I wanted to make was the way that this topic brings the whole course together.
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Kivitz, the terminus, eigenvalues and something new for stability and then something new in this expression, X transpose AX.
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Actually, that's the guide of watch in this lecture.
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So, the topic is positive definite matrix and what's my goal?
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First goal is how can I tell if a matrix is positive definite?
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So, I would like to have tests to see if you give me a 5 by 5 matrix, how do I tell if it's positive definite?
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More important is what does it mean?
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Why are we so interested in this property of positive definiteness?
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And then at the end comes some geometry. Ellipses are connected with positive definite things.
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Hyperbolas are not connected with positive definite things. So, there's a geometry too.
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But mostly it's linear algebra and this application of how do you recognize them when you have a minimum is pretty neat.
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Okay, I'm going to begin with 2 by 2. All matrices are symmetric, right?
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That's understood. The matrix is symmetric. Now, my question is, is it positive definite?
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Now, here are some, each one of these is a complete test for positive definiteness.
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If I know the eigenvalues, my test is, are they positive? Are they all positive?
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If I know the, so A is really, I look at that number A here as the one by one determinant and here's the 2 by 2 determinant.
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So, this is the determinant test. This is the eigenvalue test. This is the determinant test.
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Are the determinants growing of all N sort of, could I call them leading sub matrices?
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They're the first ones. They're the northwest Seattle sub matrices coming down from there.
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They all, all those determinants have to be positive. And then another test is the pivots.
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The pivots of a 2 by 2 matrix are that number A for sure. And since the product is the determinant,
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the second pivot must be the determinant divided by A. And then in here is going to come my favorite and my new idea,
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the one to catch about X transpose AX being positive. But we'll have to look at this guy.
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This gets like a star because for most presentations, the definition of positive definiteness would be this number 4,
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and these numbers 1, 2, 3 would be test sport. Okay, maybe I'll chuck this wire in a little bit.
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Okay, so I'll have to look at this X transpose AX. Can we just be sure, how do we know that the eigenvalue test
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and the determinant test pick out the same matrices? And let me, let's just do a few examples.
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So examples. Let me pick the matrix 2, 6, 6. Tell me, what number do I have to put there for the matrix to be positive definite?
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Tell me a sufficiently large number that would make it positive definite. Just let's just practice with these conditions in the 2 by 2 case.
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Now, when I ask you that, you don't want to find the eigenvalues, you would use the determinant test for that.
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So the first or the pivot test, that guy is certainly positive, that had to happen. And it's okay.
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How large a number here, the number had better be more than what? More than 18, right? Because if it's 8, no. More than what?
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19, is it? If I have a 19 here, is that positive definite?
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Yeah, because I get 38 minus 36. That's okay. If I had an 18, let me play it really close. If I have an 18 there, then am I positive definite?
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Not quite. I would call this guy positive, so it's, it's usual just to see that this is the borderline.
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That matrix is on the borderline. I would call that matrix positive semi definite.
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And what are the eigenvalues of that matrix? Since we're good at eigenvalues of 2 by 2's. When it's semi definite, but not definite, then the, I'm squeezing this eigenvalue test down.
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What's the eigenvalue that I know this matrix has? What kind of a matrix have I got here? The singular matrix. One of its eigenvalues is 0.
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That has an eigenvalue 0. And the other eigenvalue is from the trace 20. Okay. So that matrix has eigenvalues greater than or equal to 0.
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And it's that equal to that brought this word semi definite in. And what are the pivots of that matrix?
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So the pivots, so the eigenvalues are 0 and 20. The pivots are, well, the pivot is 2. And what's the next pivot? There isn't one.
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There we got a singular matrix here. It'll only have one pivot. You see that that's a rank 1 matrix. 2, 6 is a 6, 18 is a multiple of 2, 6. The matrix is singular. It only has 1 pivot.
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So the pivot test doesn't quite pass. Let me do the x transpose a x. Now this is the novelty now. Okay.
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What am I looking at when I look at that this new combination x transpose a x. X is any vector now. So let me compute. So any vector. Let me call its components x1 and x2.
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So that's x. And I put in here a. Let's use this example 266, 18. And here's x transpose. So x1, x2.
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We're back to real matrices after that last lecture that said what to do in the complex case. Let's come back to real matrices. So here's x transpose a x. And what I'm interested in is.
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What do I get if I multiply those together? I get some function of x1 and x2. And what is it?
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Let's see if I do this multiplication should I do it. Let me just do it slowly x1 x2 if I multiply that matrix. This is 2 x1 and 6x2.
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And the next row is 6x1 and 18x2. And now I do this final step. And what do I have? I've got 2x1 squared. That 2x1 squared is coming from this 2.
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I've got this one gives me 18. Well, so I do the ones in the middle. How many how many x1 x2s do I have?
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Let's see x1 times that 6x2 will be 6 of them. And then x2 times this one will be 6 more. I get 12. So in here is going. This is the this is the number a.
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This is the number 2b. And in here is going to the x2 times 18x2 will be 18x2 squared. And this is the number c. So it's a x1. It's like a x squared.
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2bxy and cys squared. So my first point that I wanted to make by just doing out a multiplication is that as soon as you give me the matrix.
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As soon as you give me the matrix. I can I those are the numbers that appear in the I'll call this thing a quadratic. You see it's not linear anymore.
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A x is linear. But now I've got an x transpose coming in. I'm up to degree two. Up to degree two. Maybe quadratic is the usual word a quadratic form.
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It's it's purely degree two. There's no linear part. There's no constant part. There certainly aren't any cubes or fourth powers. It's all second degree.
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And my question is is it part is that quantity positive or not? That's my it for every x1 and x2. That is my new definition. That's my definition of a positive definite matrix.
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If this quantity is positive if if if it's positive for all x is and wise all x1 x2s then I call them and then that's the matrix is positive definite.
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Now is this guy passing our test? Well we had we anticipated the answer here by by asking first about eigenvalues and pivots and what happened.
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It failed. It barely failed. If I had made this 18 down to a seven it would have totally failed.
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If I if I if I can I do that with the eraser and then I'll put back 18 because I'm seven is such a total disaster that if that well let me I'll keep seven for a second.
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Is that thing in any way positive definite? No absolutely not. I don't know what's eigenvalues but I know for sure one of them is negative.
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It's pivots are two and then the next pivot would be the determinant over two and the determinant is what what's the determinant of this thing?
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14 minus 36 I've got a determinant minus 22 the next pivot will be the pivots now of this thing are two and minus 11 or something.
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Their product being minus 22 the determinant this thing is not positive definite what would be what let me look at the X transpose A X for this guy what's if I do out this multiplication this 18 is temporarily changing to a seven.
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And I know that there's some numbers X one and X two for which that thing that function is negative and I'm desperately trying to think what they are.
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Maybe you can see can you tell me a value of X one and X two that makes this quantity negative.
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Oh maybe one and minus one. Yeah that's a that's a in this case that will work right if I took X one to be one. And X two to be minus one then I always get something positive the two and the seven minus one squared.
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But this would be minus 12 and the whole thing would be negative I'd have to minus 12 plus seven I'm negative if I drew the graph can I can I get a little picture in here if I draw the graph of this thing.
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So graphs of the function f of X Y or f of X I say or f of X Y equal this is this it's this X transpose A X this this this A X squared to be X Y and C Y squared.
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And let's take the example with these numbers.
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Okay so here's the X axis here's the Y axis and here's up is the function Z if you like or f.
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I apologize and let me just once in my life here put an arrow over these V X and so you see that that's the vector and I just gave instead of X one and X two I made them X the component X and Y.
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Okay so so what's the graph of two X squared 12 X Y and seven Y squared I'd like to see is it to get some idea actually I'm not the greatest artist but let's tell me something about this graph of this function.
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Well tell me one point it goes through the origin right even this artist can get this thing to go through the origin at when these are zero I certainly get zero.
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Okay so more points if X is one and Y is zero then I'm going upwards so I'm going up this way and I'm going up like two X squared in that direction so that that's meant to be a parabola.
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And suppose X say zero and Y increases well why could be positive or negative it's seven Y squared is this function going upward in the X direction it's going upward and in the Y direction it's going upwards and if X equals Y in the 45 degree direction it's certainly going upwards because then we'd have what about everything would be positive but.
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What what what's the graph of this function look like tell me the tell me the word that describes the graph of this function this is the non positive definite you everybody's with me here I'm for some reason got started in a negative direction here a case that isn't positive definite and what's what's the graph look like it goes up but does it.
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Do we have a minimum here does it go up from the from the origin completely no because we just checked that this thing failed it failed along the direction when X was minus Y we have a saddle point let me let me put myself let me tell you the word this thing goes up in some directions.
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But down in other directions and if we actually knew what a saddle looked like I think that will do that like am I right I mean like the way your legs go like down and the way you and looking like forward and how was that.
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And drawing the thing is even worse than describing I'm just going to say in some directions we go up and in other directions there is a saddle now I'm sorry I put that on the front board because I've no way to cover it but it's a saddle OK.
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And this is a saddle point it's a it's a point that's at the maximum in some directions and at the minimum in other directions and actually the perfect direction the look are the eigen vector directions we'll see that so this is.
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Not a positive definite matrix OK now I'm coming back to this example getting rid of this seven let's move it up to 20 let's let's make the thing really positive definite OK so this is this numbers now 20 see is now 20 OK.
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Now that passes the test which I haven't proved of course it passes the test for positive pivot.
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It passes the test for positive eigen values how can you tell that the eigen values of that matrix or positive without actually finding them of course to by two I could find them but can you see how do I know their positive I know that their product is I know that
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lambda one times lambda two is positive why because that's the determinant right lambda one times lambda two is the determinant which is 40 minus 36 is 4 so the determinant is 4 and the trace the
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value of lambda down the diagonal is 22 so they multiply to give 4 so that leaves the possibility there either both positive or both negative but if they're both negative the trace couldn't be 22 so they're both positive so both of the eigen values
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the pivots are positive the determinants are positive and I believe that this function is positive everywhere except at 0 0 of course when I write down this condition I but so I believe that X transpose let me copy X transpose A X is positive except of course at the minimum point
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at the zero vector of course I don't expect mirror so what does its graph look like and how do I check and how do I check that this really is positive so we take its graph for a minute what would be the graph of that function it does not have a saddle point let me
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go forward here and stay with this example for a while so I want to do the graph of here's my function 2 x squared 12 x y that could be positive or negative and 20 y squared
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my point is I mean you're seeing the underlying point is that the things are positive definite when when in some way these these pure squares squares we know to be positive and when those kind of overwhelmed this guy who could be positive or negative because X and
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like that same or different sign when when these are big enough they overwhelmed this guy and make the total thing positive and what would the graph now look like let me draw the X let me draw the X direction the Y direction and the origin at 0 0 I'm there where do I go as I move away from
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the origin where do I go as I move away from the origin I'm sure that I go up the origin the center point here is a minimum because this thing I believe and we better see why it's the graph is like a bowl the graph is like a bowl shape it's it's here's the minimum
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and because we've got a pure quadratic we know it sits at the origin we know it's tangent plane the first derivatives are 0 so we know we know first derivatives first derivatives are all 0 but that's not enough for a minimum
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it's first derivatives were 0 here so the the partial derivatives the first derivatives are 0 again because first derivatives are going to have an X or in a Y or a Y and then they'll be 0 at the origin it's the second derivatives that
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control everything it's the second derivatives that this matrix is telling us and somehow second so here's my point you're remembering calculus how did you decide on a minimum first requirement was that the derivative had to be 0 but then you didn't know if you had a minimum or a maximum to know that you had a minimum you had to look at the second derivative
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the second derivative had to be positive the the slope had to be increasing as you went through the minimum point the curvature had to go upwards and that's what we're doing now in two dimensions and in
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dimensions so we're doing what we did in calculus second derivative positive will now become that the matrix of second derivatives is positive definite can I just so this is like the translation of of this is how minimum coming in in in in in begin in the beginning of calculus
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the quite we had a minimum was associated with second derivative being positive and first derivative 0 of course derivative first derivative
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but it was the second derivative that told us we had a minimum and now in 1806 in linear algebra we'll have a minimum for our function now our function will have be a function not of just X but several variables the way functions really are in real life the minimum will be
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when the matrix of second derivatives the matrix here was one by one that was just one second derivative now we've got loss is positive definite
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so positive for a number translates into positive definite for a matrix and it just brings everything together you check you check pivots you check determinants you check eigenvalues or you check this minimum stuff
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okay let me come back to this graph that graph goes upwards and I have to see why how do I know that this that this function is always positive can you look at that until that it's always positive maybe two by two you could feel pretty sure but the what's the good way to show that this thing is always positive
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if we can express it as in terms of squares because that's what we know for any extent why whatever if we're squaring something we certainly are not negative
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so I believe that this expression this function could be written as a sum of squares can can you tell me see because all the problems the headaches are coming from this X Y term
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if we can get expressions if we can get that inside a square so actually what we're doing is something called that you've seen called completing the square let me let me start the square and you complete it
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okay I think we have two of X plus now I don't know how many Y's we need but you'll figure it out squared
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how many Y should I put in here to make what do I want to do the two X squares will be correct right what I want to do is put in the right number of Y's to get the 12 X Y correct
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and what is that number of Y's let's see I've got two times so I really want six X Y's to come out of here I think maybe if I put three there is that would right to you
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I have two this is we can mentally multiply that out that's X squared that's right that's 6 X Y times the two gives this one right and how many Y squares have I now got
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how many Y squares have I got from this term 18 18 was the key number remember
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now if I want to make it 20 then I've got two left two Y squares that's completed the square and it's now I can see that that function is positive because it's all squares
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I've got two squares added together I couldn't go negative what if what if I went back to that seven if instead of 20 that number was a seven then what would happen
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this would still be correct I still have this square to get the 2 X squared and the 12 X Y and I'd have 18 Y squared and then what would I do here
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I'd have to remove 11 Y squares right if I if I only had a seven here then instead of when I had a 20 I had two more to to put in
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when I had an 18 which was the marginal case I had no more to put in when I had a seven which was the case below zero the indefinite case
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I had minus 11 okay now so you can see now that this thing is a bowl it's going upwards if I cut it at a plane Z equal to one say I would get I would get a curve
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what would be the equation for that curve if I cut it at height one the equation would be this thing equal to one and that curve would be in a lips
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so actually already I've brought into the into the lecture the different pieces that we're aiming for
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we're aiming for the test which this path we're aiming for the connection to a minimum which this which we see in the graph and if we chop it at if we set this thing equal to one
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if I set that thing equal to one that what that gives me is the cross section it gives me this this curve
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and its equation is this thing equals one and that's an ellipse whereas if I cut through a saddle point I'll get a hyperbola
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okay but this this minimum stuff is is really what I'm I'm most interested in okay by I just have to ask do you recognize I mean these numbers here the two that appeared outside the three that appeared inside the
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two that appeared there actually those numbers come from elimination completing the square is our good old method of Gaussian elimination in
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express in terms of these squares the the let me show you what I mean I I just think those numbers are no accident if I take my matrix two six six and 20
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and I do elimination then the pivot is two and I take three oh yeah what's the multiplier how much of row one do I take away from row two three so what you're seeing in this completing the
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square is the pivot outside and the multiplier inside just to just do that again the the pivot is two three of three of those away from that gives me two six zero
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and what's the second pivot three of this away from this three six it will be 18 and the second pivot will also be a two so that's the that's the
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U this is the a and of course the L was one zero one and the multiplier was three so completing the square is elimination why am I happy to see
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that coming together because I know about elimination for and by and matrices I just started talking about completing the square here for two by
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two but now I see what's going on completing the square really amounts to splitting this thing into a sum of squares so what's the critical thing I have a lot of squares and inside those
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squares are multipliers but there's squares and and the question is what's outside these squares when I complete the square what are the numbers that go
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outside there's a pivot there's a pivot and that's why positive pivots give me some of squares positive pivots those pivots are the numbers that go outside the squares so
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positive pivots some of squares everything positive graph goes up a minimum at the origin it's all connected together all connected together and in the two by two case you can see those
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connections but linear algebra now can go up to three by three and by M and let's do that next can I just before I leave two by two I've written this expression matrix of second derivatives what's the matrix of second derivatives it's that's one second derivative now but if I'm in if I'm in two dimensions
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I have a two by two matrix it's the second x derivative is the second x derivative goes there the so I write it yeah I better use the this is f f x x if you like f x x that means the second derivative of f in the
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x direction f y y second derivative in the y direction those are the pure to the second derivatives they have to be positive for a minimum this number has to be positive for a minimum that number has to be positive for a minimum
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but that's not enough they those numbers have to somehow be big enough to to overcome this cross derivative why is the matrix symmetric because the second derivative of f with respect to x and y is equal to I can that's the beautiful fact about second derivatives is like to do those in either order and I get the same thing
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so this is the same as that and so that's the matrix of second derivatives and the test is it has to be positive definite you might remember from from
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a time when you talk to someone near the end of 1802 or at least in the book was the condition for a minimum for a function of two variables what when do you have a minimum for a function of two variables and believe me that's what
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the condition is first derivatives have to be zero and the matrix of second derivatives has to be positive definite so you may be remember there was an f x x times an f y y that had to be bigger than an f x y squared that's just our
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determinant to by two but now we now know the answer for three by three and by n because we can do elimination on n by n matrices we can connect to eigenvalues of n by n matrices we can do some of
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some of squares some of n squares instead of only two squares and so let's take a let me go over here to do a three by three example three by three
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examples okay oh let me make let me use shall I use my favorite matrix you've seen this matrix before let me
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let me yes let's let's use the good matrix what the one that's okay is that matrix positive definite what I'm going to ask questions about this matrix is it positive definite first of all
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what's the function associated with that matrix what's the x transpose a x is does do we have a minimum for that function at zero and then even what's the geometry
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okay first of all is the matrix positive definite given you the numbers there so you can take determinants maybe that's the quickest is that what you would do mentally if I give you a matrix on a quiz and say is it positive definite or not
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I would take that determinant and I get the answer to I would take that determinant I would get the answer for that to by two determinant I get the answer three and anybody remember the answer for the three by three determinant it was four the remember for these special matrices when we did determinants they went up two three four
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five six they just went up linearly so that matrix has the determinants are two three and four pivot what are the pivots for that matrix I'll tell you there the first pivot is two the next pivot is three over two the next pivot is four over three
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because the product of the pivots has to give me those determinants the product of these two pivots gives me that determinant the product of all the pivots gives me that determinant what are the eigenvalues
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oh I don't know yeah the eigenvalues I've got what do I have a cubic equation a degree three equation there are three eigenvalues to find
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if I believe what I've said today what do I know about these eigenvalues even though I don't know the exact numbers I think I remember the numbers because these matrices are so important that people figure them out
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but what do you what do you believe to be true about these three eigenvalues you believe that they are all positive they're all positive I think that they are two minus square root of two two and two plus the square root of two I think
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let me just I can't write those numbers down without checking the simple checks what the first simple check is the trace so if I add those numbers I get six and if I add those numbers I get six
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the other simple test is the determinant if I am I going to do this can you multiply those numbers together I guess we can multiply by two
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what's two minus square root of two times two plus square root of two that's that'll be four minus two that'll be two yeah two times two that's got the determinant right so it's got a it's got a chance of being correct and I think it is
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now what's the X transpose A X I better give myself enough room for that X transpose A X for this guy it's two X one squares and two X two squares and two X three squares those those come from the diagonal those are easy
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now off the diagonal there's a minus and a minus they come together there be a minus two and have minus two what are coming from this one two and two one position is the X one X two I'm I'm doing
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mentally a multiplication of this matrix times a row vector on the left times a column vector on the right and I know that these numbers show up in the answer the diagonal is the perfect
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squares this off diagonal is a minus two X one X two and there no X one X three and there minus two X two X three and I believe
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that that expression is always possible I believe that that curve that graph of this of this function this is my function F and I'm in more
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dimensions now than I can draw it but the graph of that function goes upwards it's a bowl or maybe the right word is is forgotten what's a long word for a bowl maybe
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paraboloid I think paraboloids comes in I'll edit the tape and get that word in bowl let's say is that so that and if I can I could complete the squares I
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could write that as a sum of three squares and those three squares would get multiplied by the three pivot and the pivots are all positive so I would have
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positive pivots time squares the net result would be a positive function and a bowl that goes upwards and then finally if I cut if I fly through this bowl if I now I'm asking
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this stretcher visualization here because I'm in four dimensions I've got X one X two X three in the base and this function is the or F or something and the its graph is going up but I'm in four
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dimensions because I've got three in the base and then the upward direction but now if I cut through this four dimensional picture at level one so suppose I cut through this thing at height one so I take all the points that are at height one
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that gives me it gave me an ellipse over there in that in that two by two case in this case this will be the equation of an ellipse
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I'll put ball in other words well not quite a football a lopsided football what what will be can I can I try to describe to you what the ellipse
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I'm sorry that the end of the matrix right at the point that let me let me be sure you see the equation two X one squared two X two squared two X three squared minus two of the cross products equal one
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that is the equation of a football so what do I mean by a football or an ellipse or I mean that well I'll draw it's like that and like that it's got a center
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and it's got it's got three principal directions this ellipse or you see what I'm saying if we had a sphere then all directions would be the thing if we had a true football or it's closer to a rugby ball really because it's more curved than a football
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it would be it would have one long direction and the other two would be equal that would be like having a matrix that had one eigenvalue repeated and then one other different so the sphere comes from like the identity matrix all eigenvalues the same
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now rugby ball comes from a case where three the three I get two of the three eigenvalues are the same but we now have a case where the typical case where the three eigenvalues are all different so this will have
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if I look at this ellipsoid correctly it will have a major axis it will have a middle axis and it will have a minor axis and those three axes will be in the direction of the eigenvectors and the length of those axes will be determined by the eigenvalues
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I can get turn this all into linear algebra because we have the right we the right thing we know about eigenvectors and eigenvalues for that matrix is what let me just tell you to repeat the main linear algebra point how could we turn what I said into into into algebra
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we would write this a as q the eigenvector matrix times lambda the eigenvalue matrix times q transpose the principal axis theorem we'll call it now the eigenvectors tell us the directions of the principal axes the eigenvalues tell us the length of those axes
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actually the length of the half length of one over the eigenvalues turns out and that is the matrix factorization which is the which is the most important matrix factorization in in our eigenvalue material so far that diagonalization for a symmetric matrix so instead of the inverse I can write the transpose
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okay I so what I tried today is to tell you the what's going on with positive definite matrices and you see all how all these pieces are there and the linear algebra connects them
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okay see on Friday