OK, this is the lecture on
positive definite matrices. I made a start on those
briefly in a previous lecture. One point I wanted to make was
the way that this topic brings the whole course together,
pivots, determinants, eigenvalues, and something
new- four plot instability and then something new in this
expression, x transpose Ax, actually that's the guy
to watch in this lecture. So, so the topic is
positive definite matrix, and what's my goal? First, first goal is, how
can I tell if a matrix is positive definite? So I would like to
have tests to see if you give me a, a
five by five matrix, how do I tell if it's
positive definite? More important is,
what does it mean? Why are we so interested
in this property of positive definiteness? And then, at the end
comes some geometry. Ellipses are connected with
positive definite things. Hyperbolas are not connected
with positive definite things, so we- it's this, we,
there's a geometry too, but mostly it's
linear algebra and -- this application of how do you
recognize 'em when you have a minim is pretty neat. OK. I'm gonna begin with two by two. All matrices are
symmetric, right? That's understood; the
matrix is symmetric, now my question is, is
it positive definite? Now, here are some -- each one of these is a complete
test for positive definiteness. If I know the eigenvalues,
my test is are they positive? Are they all positive? If I know these -- so, A is really -- I look at that number A,
here, as the, as the one by one determinant, and here's
the two by two determinant. So this is the determinant test. This is the eigenvalue test,
this is the determinant test. Are the determinants growing in
s- of all, of all end, sort of, can I call them
leading submatrices, they're the first
ones the northwest, Seattle submatrices coming
down from from there, they all, all those determinants
have to be positive, and then another
test is the pivots. The pivots of a
two by two matrix are the number A for sure,
and, since the product is the determinant,
the second pivot must be the determinant
divided by A. And then in here is gonna come
my favorite and my new idea, the, the, the the one to
catch, about x transpose Ax being positive. But we'll have to
look at this guy. This gets, like a star, because
for most, presentations, the definition of
positive definiteness would be this number four and
these numbers one two three would be test four. OK. Maybe I'll tuck this,
where, you know, OK. So I'll have to look
at this x transpose Ax. Can you, can we
just be sure, how do we know that the eigenvalue
test and the determinant test, pick out the same
matrices, and let me, let's just do a few examples. Some examples. Let me pick the matrix
two, six, six, tell me, what number do I have to
put there for the matrix to be positive definite? Tell me a sufficiently
large number that would make it
positive definite? Let's just practice with
these conditions in the two by two case. Now, when I ask
you that, you don't wanna find the eigenvalues, you
would use the determinant test for that, so, the first
or the pivot test, that, that guy is certainly
positive, that had to happen, and it's OK. How large a number here -- the
number had better be more than what? More than eighteen, right,
because if it's eight -- no. More than what? Nineteen, is it? If I have a nineteen here,
is that positive definite? I get thirty eight minus
thirty six, that's OK. If I had an eighteen, let
me play it really close. If I have an eighteen there,
then I positive definite? Not quite. I would call this
guy positive, so it's useful just to see that
this the borderline. That matrix is on
the borderline, I would call that matrix
positive semi-definite. And what are the
eigenvalues of that matrix, just since we're given
eigenvalues of two by twos, when it's semi-definite, but
not definite, then the -- I'm squeezing this
eigenvalue test down, -- what's the eigenvalue that
I know this matrix has? What kind of a matrix
have I got here? It's a singular matrix, one
of its eigenvalues is zero. That has an eigenvalue zero,
and the other eigenvalue is -- from the trace, twenty. OK. So that, that matrix has
eigenvalues greater than or equal to zero, and it's that
"equal to" that brought this word "semi-definite" in. And, the what are the
pivots of that matrix? So the pivots, so
the eigenvalues are zero and twenty, the pivots
are, well, the pivot is two, and what's the next pivot? There isn't one. We got a singular matrix here,
it'll only have one pivot. You see that that's a rank
one matrix, two six is a -- six eighteen is a
multiple of two six, the matrix is singular
it only has one pivot, so the pivot test doesn't quite The -- let me do
the x transpose Ax. pass. Now this is -- the novelty now. OK. What I looking at when I
look at this new combination, x transpose Ax. x is any vector now, so
lemme compute, so any vector, lemme call its components
x1 and x2, so that's x. And I put in here A. Let's, let's use this example
two six, six eighteen, and here's x
transposed, so x1 x2. We're back to real matrices,
after that last lecture that- that said what to do
in the complex case, let's come back to real matrices. So here's x transpose Ax,
and what I'm interested in is, what do I get if I
multiply those together? I get some function of x1
and x2, and what is it? Let's see, if I do this
multiplication, so I do it, lemme, just, I'll just
do it slowly, x1, x2, if I multiply that matrix,
this is 2x1, and 6x2s, and the next row
is 6x1s and 18x2s, and now I do this final
step and what do I have? I've got 2x1 squareds,
got 2X1 squareds is coming from this two, I've
got this one gives me eighteen, well, shall I do the
ones in the middle? How many x1 x2s do I have? Let's see, x1 times that
6x2 would be six of 'em, and then x2 times this one
will be six more, I get twelve. So, in here is going, this
is the number a, this is the number 2b, and in here is -- x2 times eighteen x2 will be
eighteen x2 squareds and this is the number c. So it's ax1 -- it's
like ax squared. 2bxy and cy squared. For my first point that I
wanted to make by just doing out a multiplication is, that is as
soon as you give me the matrix, as soon as you give me
the matrix, I can -- those are the numbers
that appear in -- I'll call this
thing a quadratic, you see, it's not
linear anymore. Ax 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 -- use the word. A quadratic form. It's purely degree two,
there's no linear part, there's no constant
part, there certainly no cubes or fourth powers,
it's all second degree. And my question is -- is that quantity
positive or not? That's -- for every x1 and x2,
that is my new definition -- that's my definition of a
positive definite matrix. If this quantity is positive,
if, if, if, it's positive for all x's and y's, all
x1 x2s, then I call them -- then that's the matrix
is positive definite. Now, is this guy
passing our test? Well we have, we anticipated
the answer here by, by asking first about
eigenvalues and pivots, and what happened? It failed. It barely failed. If I had made this
eighteen down to a seven, it would've totally failed. I do that with the eraser, and
then I'll put back eighteen, because, seven is such a
total disaster, but if -- I'll keep seven for a second. Is that thing in any
way positive definite? No, absolutely not. I don't know its eigenvalues,
but I know for sure one of them is negative. Its 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? Fourteen minus
thirty six, I've got a determinant minus twenty two. The next pivot will
be -- the pivots now, of this thing are two and
minus eleven or something. Their product being minus
twenty two the determinant. This thing is not
positive definite. What would be -- let me look
at the x transpose Ax for this guy. What's -- if I do out
this multiplication, this eighteen is temporarily
changing to a seven. This eighteen is temporarily
changing to a seven, and I know that there's
some numbers x1 and x2 for which that thing, that
function, is negative. And I'm desperately trying
to think what they are. Maybe you can see. Can you tell me a
value of x1 and x2 that makes this
quantity negative? Oh, maybe one and minus one? Yes, that's -- in this case,
that, will work, right, if I took x1 to be one,
and x2 to be minus one, then I always get something
positive, the two, and the seven minus one squared,
but this would be minus twelve and the whole thing
would be negative; I would have two minus twelve
plus seven, a negative. If I drew the graph, can I
get the little picture in here? If I draw the graph
of this thing? So, graphs, of the
function f(x,y), or f(x), so I say here f(x,y) equal
this -- x transpose Ax, this, this this ax squared,
2bxy, and cy squared. And, let's take the
example, with these numbers. OK, so here's the x axis,
here's the y axis, and here's -- up is the function. z, if you like, or f. I apologize, and let me,
just once in my life here, put an arrow over these,
these, axes so you see them. That's the vector and I just,
see, instead of x1 and x2, I made them x- the
components x and y. OK. So, so, what's a graph of 2x
squared, twelve xy, and seven y squared? I'd like to see -- I not the greatest
artist, but let's -- tell me something about
this graph of this function. Whoa, tell me one point
that it goes through. The origin. Right? Even this artist can get this
thing to go through the origin, when these are zero, I,
I certainly get zero. OK. Some 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, I'm going up, like, two x
squared in that direction. So -- that's meant
to be a parabola. And, suppose x stays
zero and y increases. Well, y 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 then the forty-five degree
direction is certainly going upwards; because
then we'd have what, about, everything would
be positive, but what? This function -- what's
the graph of this function? Look like? Tell me the word that describes
the graph of this function. This is the non-positive
definite here, everybody's with me
here, for some reason got started in a negative
direction, your case that isn't positive definite. And what's the graph look like
that goes up, but does it -- do we have a minimum here, does
it go from, 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 put myself, let me, to the least, tell you the word. This thing, goes up
in some directions, but down in other directions,
and if we actually knew what a saddle looked like or
thinks saddles do that -- the way your legs go is,
like, down, up, the way, you, looking like, forward, and, the,
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, you have no way to cover
it, but it's a saddle. OK. And, and this is a
saddle point, it's the, it's the point that's at
the maximum in some directions and at the minimum
in other directions. And actually, the perfect
directions to look are the eigenvector directions. We'll see that. So this is, 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 twenty. Let's, let's let's make the
thing really positive definite. OK. So this is, this
number's now twenty. c is now twenty. OK. Now that passes the test, which
I haven't proved, of course, it passes the test
for positive pivots. It passes the test for
positive eigenvalues. How can you tell that the
eigenvalues of that matrix are positive without
actually finding them? Of course, two by two I could
find them, but can you see -- how do I know they're positive? I know that their product is -- I know that lambda one times
lambda two is positive, why? Because that's the
determinant, right, lambda one times lambda two is
the determinant, which is forty minus thirty-six is four. So the determinant is four. And the trace, the sum down
the diagonal, is twenty-two. So, they multiply to give four. So that leaves the
possibility they're either both positive or both negative. But if they're both negative,
the trace couldn't be So they're both
positive. twenty-two. So both of the eigenvalues
that are positive, both of the pivots
are positive -- the determinants are positive,
and I believe that this function is positive everywhere
except at zero, zero, of course. When I write down
this condition, So I believe that
x transposed, let me copy, x transpose Ax is
positive, except, of course, at the minimum point,
at zero, of course, I don't expect miracles. 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 it's
graph for a minute. What would be the graph
of that function -- it does not have a saddle point. Let me -- I'll raise
the board, here, and stay with this
example for a while. So I want to do the graph
of -- here's my function, two x squared, twelve xy-s, that
could be positive or negative, and twenty y squared. But my point is, so you're
seeing the underlying point is, that, the things are
positive definite when in some way, these,
these pure squares, squares we know to be positive, and
when those kind of overwhelm this guy, who could be
m- positive or negative, because some like
or have same or have same or different signs,
when these are big enough they overwhelm this guy and
make the total thing positive, and what would the
graph now look like? Let me draw the x - well, let
me draw the x direction, the y direction, and the origin,
at zero, zero, I'm there, where do I go as I move
away from 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 minim 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 -- here's the minimum. 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 zero,
so, we know, first derivatives, First derivatives are
all zero, but that's not enough for a minimum. It's first derivatives
were zero here. So, the partial derivatives,
the first derivatives, are zero. Again, because first derivatives
are gonna have an x or an a y, or a y in them, they'll
be zero at the origin. It's the second derivatives
that control everything. It's the second derivatives
that this matrix is telling us, and somehow -- here's my point. You remember in Calculus, how
did you decide on a minimum? First requirement was, that
the derivative had to be zero. 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. The second derivative
had to be positive, 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 n dimensions. So we're doing what
we did in Calculus. Second derivative
positive, m- will now become that the matrix
of second derivatives is positive definite. Can I just -- like a translation of -- this is how minimum are coming
in, ithe beginning of Calculus -- we had a minimum was associated
with second derivative, being positive. And first derivative
zero, of course. Derivative, first
derivative, but it was the second derivative
that told us we had a minimum. And now, in 18.06,
in linear algebra, we'll have a minim
for our function now, our function will have, for
your function be a function not of just x but several variables,
the way functions really are in real life,
the minimum will be when the matrix of second
derivatives, the matrix here was one by one, there was
just one second derivative, now we've got lots. Is positive definite. So positive for a
number translates into positive
definite for a matrix. And it this brings
everything you check pivots, you check determinants,
you check all your values, or you check this minimum stuff. OK. Let me come back to this graph. That graph goes upwards. And I'll have to see why. How do I know that this,
that this function is always positive? Can you look at that and tell
that it's always positive? Maybe two by two, you
could feel pretty sure, but what's the good way to
show that this thing is always If we can express it, as, in
terms of squares, positive? because that's what we
know for any x and y, whatever, if we're
squaring something we certainly are not negative. So I believe that this
expression, this function, could be written as
a sum of squares. Can you tell me -- see, because all the
problems, the headaches are coming from this xy term. 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 start the square
and you complete it. OK, I think we
have two of x plus, now I don't remember how
many y-s we need, but you'll figure it out, squared. How many y-s should I
put in here, to make -- what do I want to do, the two
x squared-s will be correct, right? What I want to do is put
in the right number of y-s to get the twelve xy correct. And what is that number of y-s? Let's see, I've got
two times, and so I really want six xy-s
to come out of here, I think maybe if
I put three there, does that look right to you? I have two- this is, we
can mentally, multiply out, that's X squared,
that's right, that's six X Y, times the
two gives from, right, and how many Y squareds
have I now got? How many Y squareds have
I now got from this term? Eighteen. Eighteen was the key
number, remember? Now if I want to make it
twenty, then I've got two left. Two y squared-s. That's completing the
square, and it's, now I can see that that
function is positive, because it's all squares. I've got two squares,
added together, I couldn't go negative. What if I went
back to that seven? If instead of twenty that
number was a seven, then what would happen? This would still be correct,
I'd still have this square, to get the two x squared
and the twelve xy, and I'd have eighteen y squared
and then what would I do here? I'd have to remove eleven
y squared-s, right, if I only had a seven
here, then instead of -- when I had a twenty I had two
more to put in, when I had an eighteen, 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,
I had minus eleven. Now, so, you can see now,
that this thing is a bowl. OK. It's going upwards, if I cut
it at a plane, z equal to one, say, I would get, I would
get a curve, 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 an ellipse. So actually,
already, I've blocked into the lecture, the different
pieces that we're aiming for. We're aiming for the
tests, which this passed; we're aiming for the connection
to a minimum, which this -- which we see in the graph,
and if we chop it up, if we set this
thing equal to one, if I set that thing
equal to one, that -- what that gives me
is, the cross-section. It gives me this, this
curve, and its equation is this thing equals one,
and that's an ellipse. Whereas if I cut
through a saddle point, I get a hyperbola. But this minimum stuff is
really what I'm most interested OK. in. OK. 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 two that appeared there -- actually, those numbers
come from elimination. Completing the square
is our good old method of Gaussian elimination,
in expressed in terms of these squares. The -- let me show
you what I mean. I just think those
numbers are no accident, If I take my matrix two,
six, six, and twenty, and I do elimination,
then the pivot is two and I take three,
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 square, is the pivots outside and
the multiplier inside. Just do that again? The pivot is two, three -- three
of those away from that gives me two, six, zero, and
what's the second pivot? Three of this away from this,
three sixes'll be eighteen, and the second
pivot will also be a two. So that's the 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 I happy to see, happy
to see that coming together? Because I know about
elimination for m by m matrices. I just started talking about
completing the square, here, for two by twos. 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 squares are multipliers but
they're squares, and the question is, what's
outside these squares? When I complete the square,
what are the numbers that go outside? They're the pivots. They're the pivots, and that's
why positive pivots give me sum of squares. Positive pivots, those
pivots are the numbers that go outside the
squares, so positive pivots, sum 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 connections, but linear algebra now can go
up to three by three, m by m. 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? That's one second
derivative now, but if I'm in two dimensions,
I have a two by two matrix, it's the second x derivative,
the second x derivative goes there -- shall I write it --
fxx, if you like, fxx, that means the second derivative
of f in the x direction. fyy, second derivative
in the y direction. Those are the pure derivatives,
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. But, that's not enough. Those numbers have to
somehow be big enough to overcome this
cross-derivative, Why is the matrix symmetric? Because the second derivative
f with respect to x and y is equal to -- I can, that's the beautiful fact
about second derivatives, is I can do those in either order
and I get the same thing. 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, tucked in somewhere near the end of eighteen o'
two or at least in the book, was the condition for a
minimum, For a function of two variables. Let's -- when do
you have a minimum? For a function of two
variables, believe me, that's what Calculus is for. The condition is first
derivatives have to be zero. And the matrix of
second derivatives has to be positive definite. So you maybe remember there
was an fxx times an fyy that had to be bigger than
an an fxy squared, that's just our
determinant, two by two. But now, we now know the
answer for three by three, m by m, because we can do
elimination by m by m matrices, we can connect eigenvalues
of m by m matrices, we can do sum of squares, sum
of m squares instead of only two squares; and so
let's take a, let me go over here to do a
three by three example. So, three by three example. OK. Oh, let me -- shall I use my favorite matrix? You've seen this matrix before. Yes, let's use the good matrix,
four by one, oops, open. Is that matrix
positive definite? What's -- so I'm going to ask
questions about this matrix, is it positive
definite, first of all? What's the function
associated with that matrix, what's the x transpose Ax? Is -- do we have a
minimum for that function, at zero? And then even
what's the geometry? OK. First of all, is the
matrix positive definite, now I've given you the
numbers there so you can take the determinants, maybe
that's the quickest, is that what you
would do mentally, if I give you all a
matrix on a quiz and say is it positive definite or not? I would take that determinant
and I'd give the answer two. I would take that
determinant and I would give the answer for
that two by two determinant, I'd give the answer
three, and anybody remember the answer for the
three by three determinant? It was four, you remember
for these special matrices, when we do determinants, they
went up two, three, four, five, six, they just went up linearly. So that matrix has -- the
determinants are two, three, and four. Pivots. What are the pivots
for that matrix? I'll tell you, they're --
the first pivot is two, the next pivot is
three over two, the next pivot is
four over three. 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? Oh, I don't know. The eigenvalues I've got, what
do I have a cubic equation -- a degree three equation? There are three
eigenvalues to find. 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 -- I remember the numbers. Because these matrices
are so important that people figure them. But -- 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. 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. The other simple test is
the determinant, if I -- can you do this, can you
multiply those numbers together? I guess we can multiply
by two out there. What's two minus square
root of two times two plus square root of two,
that'll be four minus two, that'll be two,
yeah, two times two, that's got the determinant,
right, so it's got, it's got a chance of being
correct and I think it is. Now, what's the x transpose Ax? I better give myself
enough room for that. x transpose Ax for this guy. It's two x1 squareds, and two x2
squareds, and two x3 squareds. Those come from the
diagonal, those were easy. Now off the diagonal
there's a minus and a minus, they come together there'll be
a minus two minus two whats? Are coming from this one two and
two one position, is the x1 x2. I'm doing 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 square, this off diagonal is
a minus two x1 x2, and there are no x1 x3-s, and
there're minus two x2 x3-s. And I believe that that
expression is always positive. I believe that that
curve, that graph, really, of that function,
this is my function f, and I'm in more 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 -- just forgot, what's
a long word for bowl? Hm, maybe paraboloid, I
think, paraboloid 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 could write that as the sum of three squares,
and those three squares would get multiplied
by the three pivots. And the pivots are all positive. So I would have positive
pivots times squares, the net result would
be a positive function and a bowl which goes upwards. And then, finally, if I cut --
if I slice through this bowl, if I -- now I'm asking you
to stretch your visualization here, because I'm
in four dimensions, I've got x1 x2 x3 in the base,
and this function is z, or f, or something. And its graph is going up. But I'm in four 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. That gives me -- it gave me an ellipse over
there, in that two by two case, in this case, this will be
the equation of an ellipsoid, a football in other words. Well, not quite a football. A lopsided football. What would be, can I try
to describe to you what the ellipsoid will look
like, this ellipsoid, I'm sorry that the, that I've
ended the matrix right -- at the point, let's -- let me be
sure you've seen the equation. Two x1 squared, two x2 squared,
two x3 squared, minus two of the cross parts, equal what? That is the equation
of a football, so what do I mean by a football
or an ellipsoid? I mean that, well,
I'll draw a few. It's like that,
it's got a center, and it's got it's got
three principal directions. This ellipsoid. So -- you see what I'm saying,
if we have a sphere then all directions would be the same. If we had a true football, or
it's closer to a rugby ball, really, because it's more
curved than a football, 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 this sphere comes from,
like, the identity matrix, all eigenvalues the same. Our rugby ball comes
from a case where -- three, the three, two of the
three eigenvalues are the same. But we know how the case
where -- the typical case, where the three eigenvalues
were all different. So this will have -- How do I say it, if I look
at this ellipsoid correctly, it'll have a major axis,
it'll have a middle axis, and it'll have a minor axis. And those three axes
will be in the direction of the eigenvectors. And the lengths of
those axes will be determined by the eigenvalues. I can get -- turn this all into linear
algebra, because we have -- the right thing we know about
eigenvectors and eigenvalues, for that matrix is what? Of -- let me just tell you that,
repeat the main linear algebra point. How could we turn what
I said into algebra; we would write this A as
Q, the eigenvector matrix, times lambda, the eigenvalue
matrix times Q transposed. The principal axis
theorem, we'll call it, now. The eigenvectors tell us the
directions of the principal axes. The eigenvalues tell us the
lengths of those axes, actually the lengths, or
the half-lengths, or one over the
eigenvalues, it turns out. And that is the
matrix factorization which is the most
important matrix factorization in our
eigenvalue material so far. That's diagonalization
for a symmetric matrix, so instead of the inverse
I can write the transposed. OK. I've -- so what I've tried
today is to tell you the -- what's going on with
positive definite matrices. Ah, you see all how all
these pieces are there and linear algebra
connects them. OK. See you on Friday.
