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Ok, here we go with the quiz review
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for the third quiz that coming
on Friday
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so one key point is that the quiz
covers through Chapter 6
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Chapter 7 on linear transformation
will appear on the final exam
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but not on the quiz
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So I won't review linear
transformations today
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but they will come into the full
course review on the very last lecture
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so today I'm reviewing Chapter 6
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then I'm going to take some
of the exams
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and I'm always ready to
answer questions
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and I thought kind of helps
are memories
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if I write down the main
topics in Chapter 6
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so already on the previous quiz
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we knew how to find
the eigenvalues and eigenvectors
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Well, we knew how to find them
by that determinant of A-λI=0
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but of course it could be shortcut
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it could be like useful information
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about the eigenvalues
that we can speed things up with
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Ok, then we knew stuffs start out
with a differecial equation
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I'll do the differencial equation
problem first
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then what's special about
symmetric matrices?
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can we just say that in words?
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I'd better write it down, though
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what's special about the
symmetric matrices?
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their eigenvalues are....
real
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the eigenvalues of a symmetric
matrix always come out real
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and there are always
enough eigenvectors
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even if they are repeated eigenvalues
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there are enough eigenvectors
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and we can choose those
eigenvectors to be orthogonal
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if A=A^T, the big fact will be
that we can factor it into...
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we can diagonalize it
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and those eigenvector matrix
with eigenvectors in the column
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can be an orthogonal matrix
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so we get QλQ^T
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that's in 3 symbols
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expresses a wonderful fact
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the fundamental fact
for symmetric matrices
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Ok, then we want beyond that fact to
ask about positive definite matrices
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when the eigenvalues were positive
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I will do an example of that
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similar matrices
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now we've left symmetry
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Similar matrices are any
square matrices
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but 2 matrices are similar
if they're related that way
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and what's the key point
about similar matrices?
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somehow those matrices are
representing the same thing
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in different spaces
in Chapter 7 language
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in Chapter 6 language, what's up
with these similar matrices?
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what's the key fact...
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the key positive fact
about similar matrices
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they have the same eigenvalues
same eigenvalues
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so if one of them grows
the other one grows
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if one of them decays to 0
the other one decays to 0
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if A, that was a powers of A
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powers of A will look like powers of B
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because powers of A and powers of B
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only differ by a M^-1AM
way on the outside
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So if these are similar, then
B^k=M^-1A^kM
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and that's why I say...
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this M, it does change
the eigenvectors
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but it doesn't change the eigenvalues
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So same λs
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and then finally, I've got to
review the point about the SVD
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the Single Value Decomposition
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Ok, so that's what this quiz's
got to cover
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and now I'll just take problems
from earlier exams
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starting with the differencial equation
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Ok, and always ready for questions
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So here is an exam from
about the year zero
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and it has a 3*3
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so that was...
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but it's a pretty special-looking
matrix
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since it's got 0s on the diagonal
(see the board)
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so that's the matrix A
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Ok, step one is find the...
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Well, I want to solve that equation
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I want to find the general solution
I haven't given you a u of 0 here
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so I'm looking for the general solution
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so now, what's the form of
the general solution?
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with 3 arbitary constant's
going to be insided
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because those'll be used to
match the initial condition
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so the general form is
(see the board)
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Pure, exponential solution just
staying with that eigenvector
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of course, I haven't found yet
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the eigenvalues and the eigenvectors
that's normally the first job
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now there'll be a second one
growing like e^(λ2t)
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and a third one growing like e^(λ3t)
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so we've all done
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Well, we haven't done anything yet
actually
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I've got to find the
eigenvalues and the eigenvectors
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and then I will match u of 0
by choosing the right 3 constants
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Ok, so now I ask you about the
eigenvalues and the eigenvectors
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and you look at this matrix
and what do you see in that matrix?
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Well, I guess, of course
we might ask ourself right away
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is it singular?
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Is it singular?
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because if so, then we really
have a hat start
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we know one of the eigenvalues is zero
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is that matrix singular?
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I don't know, would you
take that determinant to find out
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or maybe you look at the first row
and third row
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and say, hey, the first row and the
third row are just opposite signs
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they are linearly denpendent
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the first column and the third
column are dependendent
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it's singular
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so one eigenvalue is 0
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let's make that λ1
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λ1=0
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Ok, now we've got a couple
of other eigenvalues of the five
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then I suppose the simplest way
is to look at A-λI
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So let me just put
(see the board)
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but actually, before I do it
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so that matrix is not
symmetric for sure, right?
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in fact it's very opposite of symmetric
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that matrix A^T
how is A^T connected to A
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it's negative
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it's an anti-symmetric matrix
skew-symmetric matrix
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and we've met
maybe a 2*2 example
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or skew-symmetric matrices
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and let me just say what's the
deal with their eigenvalues
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they are pure imaginary
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they will be on the imaginary axis
there will be some multiple of i
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if it is an anti-symmetric
skew-symmetric matrix
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so I'm looking for multiple of i
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of course, that's 0 times i
that's on the imaginary axis
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but maybe I will just do it out here
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λ^3...
(see the board)
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So I'm solving
λ^3+2λ=0
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so one root factor is out λ
and the other, the rest is λ^2+2
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Ok, this is going the way
we expect, right?
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because this gives the root λ=0
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and this gives the other 2 roots
which are λ equal...what?
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The solution of...
when this λ^2+2=0
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Then the eigenvalues
those guys are....
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what are they?
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they are multiply of i
they are just square roots of 2i
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When I said this equals to 0
I have λ^2=-2, right?
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to make that 0
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and the roots are
sqrt(2)i and -sqrt(2)i
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So now I know what those...
I'll put those in now
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e^0t is just 1
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so I don't even...
it's just 1
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this is sqrt(2)i
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and this is -sqrt(2)i
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so is the solution decay to 0?
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is it a completelely stable problem?
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where the solution is going to 0?
No
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in fact, all the things are
staying the same size
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this thing is getting multiplied by
this number e^i, something, t
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that's a number
that has a magnitude 1
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and sort of wanders around
the unit circle
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the same for this
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so that solution doesn't follow up
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and it doesn't go to zero
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Ok, and to find out what it actually is
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we'd have the plug-in
initial conditions
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but actually the next question
I ask is...
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when does the solution
return to its initial value?
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I won't even say
what is the initial value
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this is a case in which the...
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I think this solution is periodic
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after t=0
it starts that with c1, c2, c3
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and then some value of t
it comes back to that
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sort of a very special question
I'm not...It just takes 3 seconds
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because that special question
is likely beyond the quiz
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but it comes back to this start
when?
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Well, whenever we had e^2pi*i
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that's 1
and we come back again
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so it comes back to the start period...
as periodic
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when this sqrt(2)i...
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So I call it capital T for the period
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for that particular T
if that equals 2*pi*i
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then e^(this thing) is 1
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and we come around again
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so the period is...
T is determined here
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cancel the i
and T=pi*sqrt(2)
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so that's pretty neat
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we get all the information
about all solutions
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we haven't fixed on
only one particular solution
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but it comes around again
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so this is probably my first chance
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to say something about
the whole family
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of anti-symmetric or
skew-symmetric matrices
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ok, and then finally I ask...
take two eigenvectors again
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I havn't computed the eigenvectors
and it turns out they are orthogonal
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they are orthogonal
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the eigenvectors of a symmetric matrix
or skew-symmetric matrix
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are always orthognal
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I guess my conscience
makes me tell you
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what are all the matrices that
have orthogonal eigenvectors
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and symmetric is the most important class
that's the one we've spoken about
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let me just...
for that little fact down here
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orthogonal axis
eigenvectors
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A matrix has orthogonal eigenvectors
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the exact condition is quite beautiful
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that I can tell you exactly
when that happens
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it happens when
A*A^T=A^T*A
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00:13:55,980 --> 00:13:57,582
Anytime that...
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00:13:58,030 --> 00:14:01,396
that's the condition for
orthogonal eigenvectors
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00:14:02,492 --> 00:14:08,220
and because we are interested in
special families of vectors
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00:14:08,255 --> 00:14:11,403
tell me some special families that fit
203
00:14:11,691 --> 00:14:14,509
this is the whole requirement
204
00:14:14,544 --> 00:14:20,059
that's a pretty special requirement
most vector, most matrices have...
205
00:14:20,094 --> 00:14:24,622
so the average 3*3 matrix has 3
eigenvectors but not orthogonal
206
00:14:24,931 --> 00:14:28,505
but, if it happens to commute
with this transpose
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00:14:28,540 --> 00:14:33,111
then, wonderfully
the eigenvectors are orthogonal
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00:14:33,146 --> 00:14:37,543
now, do you see how symmetric
matrices pass this test
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00:14:38,378 --> 00:14:39,179
of course
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00:14:39,214 --> 00:14:43,678
if A^T=A, then both sides
are A^2, we've got it
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00:14:44,945 --> 00:14:48,180
how do anti-symmetric
matrices pass this test?
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00:14:48,616 --> 00:14:54,197
if A^T=-A
then we've got it again
213
00:14:54,232 --> 00:14:57,031
because we've got -A^2
on both sides
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00:14:57,066 --> 00:14:58,749
so those, that's another group
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00:14:59,302 --> 00:15:05,395
and finally let me ask you about another
favourite family--orthogonal matrices
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00:15:05,698 --> 00:15:09,188
do orthogonal matrices pass this test?
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00:15:09,155 --> 00:15:14,267
if A is a Q, could A pass the test
for orthogonal eigenvectors?
218
00:15:14,302 --> 00:15:18,454
Well, if A is Q, an orthogonal matrix
219
00:15:18,489 --> 00:15:20,715
what is Q^T*Q?
220
00:15:21,564 --> 00:15:22,287
It's I
221
00:15:22,322 --> 00:15:24,074
and what is Q*Q^T?
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00:15:24,420 --> 00:15:25,186
it's I
223
00:15:25,221 --> 00:15:29,288
we are talking square matrices here
so yes, it passes the test
224
00:15:29,323 --> 00:15:31,808
so the special cases are...
225
00:15:31,843 --> 00:15:39,350
symmetric/anti-symmetric(I'll say
skew-symmetric)/orthogonal
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00:15:39,748 --> 00:15:45,193
those are the 3 important classes
that are in this family
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00:15:45,228 --> 00:15:47,812
Ok, that is like a comment
228
00:15:47,847 --> 00:15:53,957
that could have been made back
in the section 6.4
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00:15:53,992 --> 00:15:59,176
Ok, I can pursue this with...
230
00:15:59,763 --> 00:16:03,499
I'd better pursue the
differential equation
231
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althogh this question didn't ask it
232
00:16:05,889 --> 00:16:12,778
to ask you to tell me how would I find
this matrix exponential e^At
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00:16:12,778 --> 00:16:16,787
So can I erase this
or I just stay with the same
234
00:16:18,822 --> 00:16:22,765
how would I find e^At
235
00:16:22,980 --> 00:16:25,943
because...
how does that come in?
236
00:16:26,289 --> 00:16:29,552
that's the key matrix for
a differential equation
237
00:16:29,552 --> 00:16:37,747
because the solution is
u(t)=e^At*u(0)
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00:16:38,264 --> 00:16:41,298
so this is like the fundamental matrix
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00:16:41,298 --> 00:16:46,211
that multiplies the...given function
and give the answer
240
00:16:47,812 --> 00:16:51,783
and how would we compute it
if we want that?
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00:16:52,985 --> 00:16:55,734
We don't always have to find e^At
242
00:16:55,769 --> 00:16:59,990
because I can go directly to the answer
without any e^At
243
00:17:00,025 --> 00:17:04,023
but hiding here isn't the e^At
and how would I compute it?
244
00:17:05,525 --> 00:17:11,637
Well, if A is diagonalizible
"if"
245
00:17:11,672 --> 00:17:15,416
so now I'm going to put in
my usual "if"
246
00:17:16,378 --> 00:17:20,469
if A can be diagonalized
247
00:17:21,805 --> 00:17:24,600
and everyone remembers that
there isn't a "if" there
248
00:17:24,894 --> 00:17:28,202
because it might not
have enough eigenvectors
249
00:17:28,237 --> 00:17:32,443
this example does have enough
random matrices have enough
250
00:17:32,774 --> 00:17:36,228
so if we can diangonalize
then we get a nice formula for this
251
00:17:36,263 --> 00:17:39,455
because an S comes way
out of the beginning
252
00:17:39,490 --> 00:17:42,507
and Inv[S] comes way out of the end
253
00:17:42,542 --> 00:17:46,471
and we only have to take
the exponential of λ
254
00:17:46,506 --> 00:17:49,827
And that's the diangonal matrix
255
00:17:49,862 --> 00:17:58,797
So that's just e^λ*AT, so these guys
are showing up now, and e^AT
256
00:17:59,553 --> 00:18:03,421
Ok, that's really the quick
review of that formula
257
00:18:05,431 --> 00:18:11,472
it's something we can compute quickly
if we have done the Sλ part
258
00:18:12,853 --> 00:18:16,887
if we know Sλ
then it's not hard to take that step
259
00:18:16,922 --> 00:18:20,393
Ok, that's some comments
on differential equations
260
00:18:20,428 --> 00:18:26,822
I'd like to go on to a next question
that I started here
261
00:18:27,849 --> 00:18:32,381
and it's got several parts
I can just read it out
262
00:18:33,162 --> 00:18:39,060
what would be given is a 3*3 matrix
and were told these eigenvalues
263
00:18:39,060 --> 00:18:43,417
except, one of these is like
we don't know
264
00:18:44,290 --> 00:18:46,640
and were told the eigenvectors
265
00:18:47,004 --> 00:18:49,704
and I want to ask you about the matrix
266
00:18:49,739 --> 00:18:52,567
ok, so, first question
267
00:18:54,001 --> 00:18:57,007
Is the matrix diagonalizable?
268
00:18:58,519 --> 00:19:04,199
and I really mean for which c
because I don't know c
269
00:19:04,234 --> 00:19:06,194
so my question will all be...
270
00:19:07,029 --> 00:19:11,856
for which, is our condition on c
does one c work?
271
00:19:11,891 --> 00:19:16,506
But your answer should tell me
all the Cs that work
272
00:19:16,541 --> 00:19:18,648
I'm not asking like...
273
00:19:18,683 --> 00:19:22,508
for you to tell me: well, C works
yea, let's check it out
274
00:19:22,543 --> 00:19:29,144
I want to know all the C's
that make it diagonalizable
275
00:19:34,443 --> 00:19:38,553
Ok, what is the c
when diagonalizable
276
00:19:38,588 --> 00:19:41,260
we need enough eigenvectors, right?
277
00:19:42,050 --> 00:19:43,895
we don't care what those
eigenvalues are
278
00:19:43,930 --> 00:19:46,571
but eigenvectors that account
for diagonalizable
279
00:19:46,606 --> 00:19:51,172
and we need 3 indenpendent ones
and are those 3 guys indenpendent?
280
00:19:51,928 --> 00:19:53,081
Yes
281
00:19:54,000 --> 00:19:56,230
actually let's look at them
for a moment
282
00:19:56,265 --> 00:19:58,815
what do you see about those 3 vectors
right away
283
00:19:59,472 --> 00:20:01,644
they are more than indenpendent
284
00:20:01,679 --> 00:20:03,131
they are...
285
00:20:04,160 --> 00:20:09,243
Can you see what,
why are those three got chosen?
286
00:20:09,345 --> 00:20:12,094
Because it'll come up
in the next parts
287
00:20:12,129 --> 00:20:15,556
they are orthgonal
288
00:20:15,556 --> 00:20:17,609
those eigenvectors are orthogonal
289
00:20:17,644 --> 00:20:24,074
they are certainly indenpendent
so the answer to diagonalizable is yes
290
00:20:24,109 --> 00:20:29,112
all c, all c
doesn't matter
291
00:20:29,147 --> 00:20:30,749
c could be a repeated guy
292
00:20:30,784 --> 00:20:33,632
we've got enough eigenvectors
so that's what we care about
293
00:20:33,667 --> 00:20:35,633
ok, second question
294
00:20:36,022 --> 00:20:38,858
for which values of C is symmetric
295
00:20:40,861 --> 00:20:41,777
Ok
296
00:20:44,930 --> 00:20:46,277
what's the answer to that one?
297
00:20:48,580 --> 00:20:51,310
if we know the same setup
298
00:20:52,080 --> 00:20:54,985
if we know that much about that
we know those eigenvectors
299
00:20:55,020 --> 00:20:57,095
and we've noticed
they're orthogonal
300
00:20:58,362 --> 00:21:00,658
then which Cs will work?
301
00:21:02,774 --> 00:21:08,634
so the eigenvalues of that
symmetric matrix have to be real
302
00:21:08,669 --> 00:21:10,957
so, all real Cs
303
00:21:11,656 --> 00:21:17,607
if C was i, the matrix wouldn't
have been symmetric
304
00:21:19,109 --> 00:21:21,061
but if C is a real number
305
00:21:21,096 --> 00:21:25,983
then we've got real eigenvalues
we've got orthogonal eigenvectors
306
00:21:26,018 --> 00:21:27,415
that matrix is symmetric
307
00:21:27,450 --> 00:21:29,060
Ok, positive definite
308
00:21:32,005 --> 00:21:33,849
Ok, so that...
309
00:21:36,975 --> 00:21:40,100
Now this is the subcase of symmetric
310
00:21:41,563 --> 00:21:45,715
so we need c to be real
so we've got a symmetric matrix
311
00:21:46,180 --> 00:21:49,455
but we also want the things
to be positive definite
312
00:21:50,093 --> 00:21:52,534
now we're looking at the eigenvalues
313
00:21:52,569 --> 00:21:55,103
we've got a lot of test
for positive definite
314
00:21:55,530 --> 00:22:00,281
but eigenvalues that we know them is
certainly a good quick clean test
315
00:22:00,980 --> 00:22:03,691
could this matrix be positive definite?
316
00:22:05,486 --> 00:22:07,232
No, no
317
00:22:07,267 --> 00:22:09,139
Because it's got the eigenvalue 0
318
00:22:10,135 --> 00:22:12,694
it could be positive, semi-definite
319
00:22:12,695 --> 00:22:15,864
you know, like consolation prize
320
00:22:15,899 --> 00:22:19,434
if c>=0
321
00:22:19,469 --> 00:22:21,748
it would be positive, semi-definite
322
00:22:21,783 --> 00:22:24,831
but it's not...
no
323
00:22:24,832 --> 00:22:26,599
semi-definite
324
00:22:26,634 --> 00:22:34,362
if I put that comment in semi-definite
that the condition would be c>=0
325
00:22:34,929 --> 00:22:36,487
and will be all right
326
00:22:36,522 --> 00:22:38,457
ok, next part
327
00:22:38,492 --> 00:22:40,399
Is it a Markov matrix?
328
00:22:44,130 --> 00:22:46,868
Is...could this matrix be...
329
00:22:46,903 --> 00:22:52,796
if I choose the number C correctly?
A Markov matrix
330
00:22:58,611 --> 00:23:01,766
well, what we know about
the Markov matrices?
331
00:23:02,868 --> 00:23:05,573
mainly, we know something
about the eigenvalues
332
00:23:05,608 --> 00:23:09,269
one eigenvalue is always 1
333
00:23:09,304 --> 00:23:14,673
and the other eigenvalues
are smaller, not larger
334
00:23:14,708 --> 00:23:17,340
so an eigenvalue two can happen
335
00:23:17,375 --> 00:23:19,005
the answer is No
336
00:23:19,040 --> 00:23:21,929
Not, never a Markov matrix
337
00:23:22,229 --> 00:23:30,345
Ok, and finally could one half of A
be a projection matrix?
338
00:23:30,380 --> 00:23:34,244
so could this be twice
the projection matrix
339
00:23:34,279 --> 00:23:35,898
so let me write this way
340
00:23:35,933 --> 00:23:40,394
Could A/2 be a projection matrix?
341
00:23:45,033 --> 00:23:47,353
Ok, what a projection matrix is?
342
00:23:47,388 --> 00:23:51,637
they are real, I mean
they're symmetric
343
00:23:51,672 --> 00:23:53,390
so their eigenvalues are real
344
00:23:53,425 --> 00:23:57,185
but more than that we know
what eigenvalues have to be
345
00:23:58,404 --> 00:24:01,551
what are the eigenvalues
of a projection matrix have to be?
346
00:24:01,552 --> 00:24:04,657
Any nice matrix...
347
00:24:04,657 --> 00:24:08,125
we've got an idea of this eigenvalue
348
00:24:08,977 --> 00:24:13,983
so the eigenvalues of projection
matrices are 0 and 1
349
00:24:14,623 --> 00:24:16,825
0 and 1, normally
350
00:24:16,860 --> 00:24:21,707
because P^2=P
let me call this matrix P
351
00:24:22,022 --> 00:24:26,952
So P^2=P
so λ^2=λ
352
00:24:26,952 --> 00:24:37,491
(see the board)
353
00:24:37,491 --> 00:24:41,809
Ok, now what values of C
will work there
354
00:24:42,641 --> 00:24:48,866
So then we need...
there're some values that will work
355
00:24:48,867 --> 00:24:50,407
then what will work?
356
00:24:51,185 --> 00:24:53,515
c=0 will work?
357
00:24:55,034 --> 00:24:56,947
or what else will work?
358
00:24:59,505 --> 00:25:02,796
c=2
359
00:25:02,796 --> 00:25:06,438
because if c=2
then when we divide by 2
360
00:25:07,199 --> 00:25:10,059
this eigenvalue of 2 will drop to 1
361
00:25:10,094 --> 00:25:11,587
and so what the other one
362
00:25:11,622 --> 00:25:14,219
so all c=2, ok
363
00:25:14,254 --> 00:25:15,729
those are the guys
that will work
364
00:25:15,764 --> 00:25:21,404
and then with the fact that
those eigenvectors were orthogonal
365
00:25:22,064 --> 00:25:26,270
the fact that those eigenvalues were
orthogonal, carries us a lot away here
366
00:25:26,305 --> 00:25:30,076
if they were orthogonal, then
symmetric would have been dead
367
00:25:30,111 --> 00:25:32,157
positive definite would have been dead
368
00:25:32,192 --> 00:25:33,693
projection would have been dead
369
00:25:35,368 --> 00:25:37,642
but those eigenvectors
were orthogonal
370
00:25:37,677 --> 00:25:40,616
so it came down to the eigenvalues
371
00:25:40,651 --> 00:25:46,360
Ok, that was like a chance to
review a lot of this chapter
372
00:25:51,381 --> 00:25:53,000
so I take these...
373
00:25:53,035 --> 00:26:03,533
so I jump to the singular value decompo-
sition end as a third topic for the review
374
00:26:04,486 --> 00:26:07,041
Ok, so I'm going to jump to this
375
00:26:14,005 --> 00:26:17,352
Ok, so this is a singular value
decomposition
376
00:26:17,984 --> 00:26:20,986
as known as, to everybody it's SVD
377
00:26:21,785 --> 00:26:32,806
that's a factorization of A into
orthogonal times diagonal times orthogonal
378
00:26:34,374 --> 00:26:41,066
and we always call those
U∑V^T
379
00:26:41,889 --> 00:26:44,582
Ok, and the key to that...
380
00:26:46,412 --> 00:26:49,165
this is for every matrix
381
00:26:49,200 --> 00:26:53,737
every A, rectangular, doesn't matter
382
00:26:53,772 --> 00:26:59,149
whatever can be this decomposition
so it's really important
383
00:26:59,793 --> 00:27:05,357
and the key to it
is to look at things like A^TA
384
00:27:05,392 --> 00:27:08,332
and we remembered what
happened with A^TA
385
00:27:08,367 --> 00:27:18,146
if I just transpose that
I get (see the board)
386
00:27:18,147 --> 00:27:20,960
and the result is
387
00:27:23,410 --> 00:27:25,356
V on the outside
388
00:27:26,304 --> 00:27:29,000
U^TU is the identity
389
00:27:29,035 --> 00:27:31,229
because it is an orthogonal matrix
390
00:27:31,697 --> 00:27:33,493
so I'm just left with
391
00:27:33,494 --> 00:27:36,454
∑^T
∑ in the middle
392
00:27:37,870 --> 00:27:42,914
as a diangonal, possibly rectangular
diagnonal by its transpose
393
00:27:42,949 --> 00:27:47,753
so the result is... this is orthogonal
diagnonal, orthogonal
394
00:27:50,827 --> 00:27:56,407
so I guess actually
this is the SVD for A^T A
395
00:27:56,442 --> 00:28:00,780
here I see, orthogonal
diagononal and orthogonal
396
00:28:00,815 --> 00:28:01,618
great
397
00:28:02,995 --> 00:28:07,510
but a little more is happening
398
00:28:07,545 --> 00:28:10,985
for A^T A, the difference....
399
00:28:12,439 --> 00:28:14,564
the orthogonal guys are the same
400
00:28:14,599 --> 00:28:16,353
These are V and V^T
401
00:28:16,388 --> 00:28:17,848
what am I seeing here?
402
00:28:17,883 --> 00:28:25,138
I'm seeing the factorization for a
symmetric matrix, this thing is symmetric
403
00:28:27,231 --> 00:28:30,928
so in the symmetric case
U is the same as V
404
00:28:31,491 --> 00:28:34,249
U is the same as V
for this symmetric matrix
405
00:28:34,250 --> 00:28:35,788
and of course, we see it happens
406
00:28:35,823 --> 00:28:40,467
Ok, so that tells us right away
what V is
407
00:28:40,502 --> 00:28:49,401
V is the eigenvector matrix
for A^T A
408
00:28:50,428 --> 00:28:58,004
Ok, now, if you're here when
I lectured about this topic
409
00:28:58,039 --> 00:29:01,522
when I gave you the lecture
on single values decompositions
410
00:29:01,557 --> 00:29:04,450
you remembered that I got into trouble
411
00:29:07,278 --> 00:29:09,033
I'm sorry to remember that myself
412
00:29:09,068 --> 00:29:10,763
but it happened
413
00:29:10,798 --> 00:29:12,892
ok, how did it happen?
414
00:29:13,742 --> 00:29:18,179
I did...I was in
great shape for a while
415
00:29:18,179 --> 00:29:21,772
so I found the eigenvectors
for A^T A, good
416
00:29:22,987 --> 00:29:25,526
I found the singular values
what were they?
417
00:29:25,561 --> 00:29:27,774
what were the singular values?
418
00:29:27,809 --> 00:29:29,561
the singular value...
419
00:29:29,561 --> 00:29:35,950
the singular value number i
for this guy and ∑
420
00:29:36,426 --> 00:29:40,541
this is diagonal with the number ∑
421
00:29:40,576 --> 00:29:45,979
This diagonal is ∑1, ∑2
up to the rank, ∑r
422
00:29:46,014 --> 00:29:47,617
those are nonzero one
423
00:29:49,778 --> 00:29:52,002
so I found those and what are they?
424
00:29:52,037 --> 00:29:53,762
remind me about that...
425
00:29:53,797 --> 00:29:57,417
well, here I'm seeing them squared
426
00:29:58,079 --> 00:30:04,069
so their squares are
the eigenvalues of A^T A
427
00:30:04,510 --> 00:30:05,256
Good
428
00:30:06,592 --> 00:30:08,631
so I just take this square root
429
00:30:08,666 --> 00:30:11,037
If I want the eigenvalues of A^T A
430
00:30:11,072 --> 00:30:13,191
if I want the ∑s and I know these
431
00:30:13,226 --> 00:30:15,543
I take the square root
the positive square root
432
00:30:15,578 --> 00:30:19,516
Ok, where did I ran into trouble?
433
00:30:21,763 --> 00:30:24,976
Well, then my final step was to find U
434
00:30:26,335 --> 00:30:28,314
and I didn't read the book
435
00:30:29,480 --> 00:30:35,182
so, I did something
that was practically right
436
00:30:35,217 --> 00:30:39,660
but, well, I get it practically right
it is not right
437
00:30:39,940 --> 00:30:44,932
Ok, so I'll look at AA^T
438
00:30:45,857 --> 00:30:48,483
what happened when I look at AA^T
439
00:30:48,518 --> 00:30:52,441
let me just put it here
and then I can see it
440
00:30:52,476 --> 00:30:55,647
Ok,so here is AA^T
441
00:30:58,188 --> 00:31:08,708
(see the board)
442
00:31:08,743 --> 00:31:11,764
and then in the middle
the identity again
443
00:31:12,588 --> 00:31:19,122
so looks great
U(∑∑^T)U^T, fine
444
00:31:19,908 --> 00:31:25,505
all good, and again
now the eigenvector...
445
00:31:25,540 --> 00:31:29,996
now these columns of U
are the eigenvectors
446
00:31:30,031 --> 00:31:32,892
U is the eigenvector
matrix for this guy
447
00:31:34,680 --> 00:31:36,639
that was correct
so I did that fine
448
00:31:38,026 --> 00:31:41,557
where did something go wrong?
assignment was wrong
449
00:31:41,854 --> 00:31:45,398
assignment was wrong because...
now I see
450
00:31:45,433 --> 00:31:47,950
actually somebody told me
right after the class
451
00:31:50,272 --> 00:31:56,674
we can tell from this description
which assigned to give the eigenvectors
452
00:31:57,586 --> 00:32:01,943
if these are the eigenvectors
of this matrix
453
00:32:01,978 --> 00:32:05,803
Well, if you give me an eigenvector
and I change all its signs
454
00:32:05,838 --> 00:32:07,942
you, we'll still get another
eigenvector
455
00:32:08,486 --> 00:32:12,769
so what I wasn't able to determine
and I have a fifty-fifty chance and...
456
00:32:12,804 --> 00:32:15,392
life let me down
457
00:32:15,427 --> 00:32:19,413
the signs I just happened to
pick for the eigenvectors
458
00:32:19,448 --> 00:32:22,597
one of them should reverse the sign
459
00:32:23,322 --> 00:32:24,833
so from this
460
00:32:24,833 --> 00:32:30,462
I can't tell whether the eigenvector
or its negative
461
00:32:30,497 --> 00:32:32,618
is the right one to use in there
462
00:32:32,653 --> 00:32:39,198
so the right way to do it is to
having settle the signs, the Vs also
463
00:32:39,975 --> 00:32:42,600
I don't know which sign to choose
but I choose one
464
00:32:43,588 --> 00:32:44,638
I choose one
465
00:32:44,673 --> 00:32:49,490
and then I should have
used instead
466
00:32:49,846 --> 00:32:54,801
I should have used the one
that tells me what sign to choose
467
00:32:54,836 --> 00:33:02,043
the rule that
(see the board)
468
00:33:03,862 --> 00:33:05,861
so having decided on the V
469
00:33:07,062 --> 00:33:08,792
I multiply by A
470
00:33:08,827 --> 00:33:11,463
and I notice the factor ∑
coming out...
471
00:33:11,498 --> 00:33:13,119
and it will be a unit vector there
472
00:33:13,154 --> 00:33:19,278
and I now know which...
exactly what it is...
473
00:33:19,313 --> 00:33:21,904
and not only up to a change of sign
474
00:33:21,939 --> 00:33:23,394
so that's the goods
475
00:33:23,429 --> 00:33:26,864
and of course, this is the
main point about the SVD
476
00:33:27,532 --> 00:33:29,966
that's the point that we diagonalize
477
00:33:30,001 --> 00:33:38,441
that A*V=U*∑
478
00:33:39,283 --> 00:33:41,037
that's the same as that
479
00:33:41,072 --> 00:33:43,634
Ok, so that's like correcting...
480
00:33:46,359 --> 00:33:51,068
the wrong sign
from that earlier lecture
481
00:33:51,600 --> 00:33:55,536
and that was complete, so that tell you
how to compute the SVD
482
00:33:56,241 --> 00:34:00,082
now, on the quiz I'm going to ask...
well, maybe on the final
483
00:34:00,117 --> 00:34:01,958
so we've got quiz and final ahead
484
00:34:02,578 --> 00:34:08,070
sometimes you might be asked to
find the SVD, if I give you the matrix
485
00:34:08,105 --> 00:34:10,343
let me now come back to the main board
486
00:34:12,642 --> 00:34:18,840
Or I might give you the pieces
487
00:34:19,640 --> 00:34:22,570
and I might ask you
something about the matrix
488
00:34:23,387 --> 00:34:28,421
for example, suppose I ask you...
489
00:34:29,212 --> 00:34:30,662
Oh, let's say...
490
00:34:32,619 --> 00:34:35,604
Well, if I tell you what ∑ is
491
00:34:37,756 --> 00:34:40,524
Well, let's...
Ok, let's take one example
492
00:34:40,957 --> 00:34:43,092
suppose ∑ is ...
493
00:34:45,794 --> 00:34:48,305
so, all that's...
how we will compute them
494
00:34:48,340 --> 00:34:50,050
now suppose I give you this
495
00:34:50,085 --> 00:34:55,175
suppose I give you ∑ is
say, 3, 2
496
00:34:58,964 --> 00:35:03,827
and I tell you that U is...
has a couple of columns
497
00:35:04,580 --> 00:35:06,738
and V has a couple of columns
498
00:35:09,406 --> 00:35:14,670
Ok, those are orthogonal
columns of course
499
00:35:14,705 --> 00:35:16,566
because U and V are orthogonal
500
00:35:16,601 --> 00:35:20,318
I'm just sort of like getting you
to think about the SVD
501
00:35:20,353 --> 00:35:22,650
Because we only have
one lecture about it
502
00:35:22,685 --> 00:35:26,838
and one homework
and...
503
00:35:28,090 --> 00:35:30,440
what kind of a matrix have I got
504
00:35:30,475 --> 00:35:32,201
what do I know about this matrix?
505
00:35:33,859 --> 00:35:35,922
all I really know right now is that
506
00:35:35,957 --> 00:35:39,401
the singular values of ∑ are 3 and 2
507
00:35:40,083 --> 00:35:44,762
and the only thing interesting that
I can see in that is it is not zero
508
00:35:44,797 --> 00:35:49,494
I know the matrix is
non-singular, right?
509
00:35:50,267 --> 00:35:53,511
that's invertible
I don't have zero eigenvalues
510
00:35:53,546 --> 00:35:56,563
and this is singular values
that is invertable
511
00:35:56,598 --> 00:36:01,143
there is a typical SVD
for a nice 2*2
512
00:36:03,108 --> 00:36:06,471
non-singular invertible
good matrix
513
00:36:07,082 --> 00:36:09,273
if I actually gave you a matrix
514
00:36:09,308 --> 00:36:13,774
than you have to find the Us
and Vs that we just focused there
515
00:36:13,809 --> 00:36:18,019
now, what if the two
wasn't the two...but it was...
516
00:36:18,054 --> 00:36:20,680
Well, let me make it
an extreme case here
517
00:36:20,715 --> 00:36:22,325
suppose it was -5
518
00:36:25,006 --> 00:36:26,334
that's wrong
519
00:36:26,369 --> 00:36:30,346
right away, that's not a
singular value decomposition, right?
520
00:36:30,381 --> 00:36:33,016
the singular values are not negative
521
00:36:34,161 --> 00:36:38,021
so that's not a singular value
decomposition and forget it
522
00:36:38,056 --> 00:36:40,783
Ok, so let me ask you about that one
523
00:36:42,193 --> 00:36:44,602
what can you tell me about that matrix?
524
00:36:47,163 --> 00:36:48,316
it's singular, right?
525
00:36:49,667 --> 00:36:52,132
got a singular matrix there
in the middle?
526
00:36:52,934 --> 00:36:54,968
and let's see
527
00:36:55,003 --> 00:36:56,465
can you solve...
528
00:36:56,500 --> 00:37:02,709
Ok, it's singular
maybe you can tell me its rank
529
00:37:04,153 --> 00:37:05,411
what's the rank of A
530
00:37:06,097 --> 00:37:10,652
it's clearly...
somebody just say it
531
00:37:11,076 --> 00:37:13,487
1, thanks
the rank is 1
532
00:37:15,932 --> 00:37:19,500
so the null space, what's the
dimension of the null-space?
533
00:37:20,835 --> 00:37:26,097
1, right?
we've got a 2*2 matrix of rank 1
534
00:37:26,132 --> 00:37:30,461
so all that starts from the beginning
of the course is still with us
535
00:37:32,700 --> 00:37:36,883
the dimension of those fundamental
spaces is still central
536
00:37:36,918 --> 00:37:38,772
and a basis for them
537
00:37:38,773 --> 00:37:41,986
Now can you tell me a vector
that's in the null space?
538
00:37:42,872 --> 00:37:48,727
that would be my last point
to make about the SVD
539
00:37:49,340 --> 00:37:52,023
can you tell me a vector
that's in the null space
540
00:37:54,978 --> 00:37:57,583
so it's really, it's somehow...
541
00:37:57,583 --> 00:38:00,970
what would I multiply by
and get zero here?
542
00:38:03,252 --> 00:38:05,822
I think the answer is probably v2
543
00:38:06,929 --> 00:38:10,346
I think probably v2
is in the null space
544
00:38:10,381 --> 00:38:17,322
I think that must be the eigenvector
going with this zero eigenvalue
545
00:38:18,446 --> 00:38:19,704
I will look at that
546
00:38:19,739 --> 00:38:23,880
and I could ask you about
the null space of A^T
547
00:38:23,915 --> 00:38:26,703
I could ask you the column space
All that space
548
00:38:26,703 --> 00:38:29,391
everything is sitting there of the SVD
549
00:38:29,426 --> 00:38:32,441
SVD takes a little more
time to compute
550
00:38:32,476 --> 00:38:37,998
but it displays all the good
stuff about the matrix
551
00:38:37,998 --> 00:38:41,777
Ok, any question about the SVD
552
00:38:42,585 --> 00:38:50,376
let me keep going with
further topics
553
00:38:50,411 --> 00:38:51,282
Now let's see
554
00:38:51,317 --> 00:38:53,523
similar matrices we've talked about...
555
00:38:53,558 --> 00:38:57,278
let me see if I've got another...
556
00:38:57,278 --> 00:39:06,426
Ok, here is a true-false
so we can do that easily
557
00:39:06,853 --> 00:39:10,766
so, question, A, given...
558
00:39:12,579 --> 00:39:20,979
A is symmetric and orthogonal
559
00:39:23,423 --> 00:39:24,154
Ok
560
00:39:29,295 --> 00:39:32,307
so beautiful matrices I get
don't come alone every day
561
00:39:32,342 --> 00:39:37,835
but, what can we say first
about its eigenvalues?
562
00:39:38,654 --> 00:39:41,940
actually, of course
we've got...
563
00:39:41,940 --> 00:39:44,457
here are two most important
classes of matrices
564
00:39:44,492 --> 00:39:46,403
that we're looking at the intersection
565
00:39:48,130 --> 00:39:51,160
so those really are
neat matrice
566
00:39:51,195 --> 00:39:53,077
what can you tell me about the...
567
00:39:53,077 --> 00:39:55,192
what could the possible eigenvalues be?
568
00:39:55,227 --> 00:40:00,423
eigenvalues can be what?
569
00:40:00,458 --> 00:40:03,657
what do I know about the eigenvalues
of the symmetric matrix?
570
00:40:03,692 --> 00:40:06,920
λ is real
571
00:40:06,955 --> 00:40:10,864
what do I know about the eigenvalues
of orthogonal matrix?
572
00:40:12,831 --> 00:40:15,928
Ha, maybe nothing
573
00:40:17,889 --> 00:40:21,137
what do I know about the
eigenvalues of orthogonal matrix?
574
00:40:21,172 --> 00:40:22,629
so what feels right?
575
00:40:24,765 --> 00:40:29,217
[not clear]
576
00:40:29,217 --> 00:40:36,243
the eigenvalues of the orthogonal
matrix also have the magnitude 1
577
00:40:36,278 --> 00:40:39,066
orthogonal matrices are like rotation
578
00:40:39,101 --> 00:40:40,837
they are not changing the length
579
00:40:41,241 --> 00:40:44,164
so orthogonal...
the eigenvalues are 1
580
00:40:44,199 --> 00:40:45,905
let me just show you why
581
00:40:49,819 --> 00:40:50,649
why?
582
00:40:52,935 --> 00:40:54,835
so the matrix, let's call it...
583
00:40:54,870 --> 00:40:58,067
can I call it Q for orthogonal
for the moment?
584
00:40:58,067 --> 00:41:01,000
And if I look at Qx=λx
585
00:41:01,035 --> 00:41:04,981
how do I see that
this thing has magnitude 1
586
00:41:06,360 --> 00:41:08,325
I take the length of both sides
587
00:41:08,360 --> 00:41:14,215
this is taking length
this is whatever magnitude is
588
00:41:14,250 --> 00:41:15,674
times the length of X
589
00:41:15,709 --> 00:41:18,302
and what's the length of Qx?
590
00:41:18,337 --> 00:41:20,525
Q is an orthogonal matrix
591
00:41:20,560 --> 00:41:22,600
this is something you should know
592
00:41:23,601 --> 00:41:25,505
it's the same as the length of x
593
00:41:25,540 --> 00:41:28,773
orthogonal matrices don't change length
594
00:41:28,808 --> 00:41:32,921
so λ has to be 1
595
00:41:32,956 --> 00:41:34,602
right, ok
596
00:41:34,637 --> 00:41:38,833
that words committing to memory
that could show up again
597
00:41:39,587 --> 00:41:43,590
Ok, so, what's the answer now
to this question?
598
00:41:43,625 --> 00:41:45,305
what can the eigenvalues be?
599
00:41:45,340 --> 00:41:47,540
there are only 2 possibilities
600
00:41:47,575 --> 00:41:51,604
and they are 1 and...
601
00:41:55,844 --> 00:42:00,979
the other possibility is -1, right
602
00:42:01,712 --> 00:42:04,736
Because these have the right magnitude
and they are real
603
00:42:04,771 --> 00:42:09,011
ok, ok
true...ok
604
00:42:09,945 --> 00:42:10,875
true or false?
605
00:42:10,910 --> 00:42:13,417
A is sure to be positive definite
606
00:42:15,154 --> 00:42:16,540
Well, it is a great matrix
607
00:42:16,575 --> 00:42:18,620
but is it sure to be positive definite?
608
00:42:19,411 --> 00:42:22,392
No, if it could have
an eigenvalue of -1
609
00:42:22,427 --> 00:42:23,863
it wouldn't be positive definite
610
00:42:24,526 --> 00:42:27,710
true or false, it has no
repeated eigenvalues
611
00:42:30,631 --> 00:42:32,030
that's false, too
612
00:42:32,714 --> 00:42:36,504
in fact it's going to have repeated
eigenvalues if it is a biggest 3*3
613
00:42:36,539 --> 00:42:39,957
one of this has to be repeated
614
00:42:39,992 --> 00:42:42,609
sure, so it has got
repeated eigenvalues
615
00:42:42,644 --> 00:42:44,673
but, is it diagonalizable?
616
00:42:46,076 --> 00:42:48,374
it's got many many
repeated eigenvalues
617
00:42:48,409 --> 00:42:51,288
50*50 has certainly got
a lot of repeatations
618
00:42:51,323 --> 00:42:53,393
is it diagonalizable?
619
00:42:54,156 --> 00:42:56,841
yes, all symmetric matrices...
620
00:42:56,876 --> 00:43:00,055
all orthogonal matrices
can be diagonalized
621
00:43:00,090 --> 00:43:05,577
and in fact, the eigenvectors can
even be chosen orthogonal
622
00:43:05,612 --> 00:43:11,854
so it can be diagonalized in the
best way with Q not just any all this
623
00:43:11,854 --> 00:43:14,690
ok
Is it non-singular?
624
00:43:15,462 --> 00:43:19,037
Is this symmetric orthogonal
matrix non-singular?
625
00:43:21,334 --> 00:43:25,140
sure, orthogonal matrices are
always non-singualr
626
00:43:25,768 --> 00:43:28,892
and obviously, we don't have
any zero eigenvalues
627
00:43:29,581 --> 00:43:31,753
it is sure to be diagonalizable?
628
00:43:31,788 --> 00:43:35,824
yes, prove that...
now here is the final step
629
00:43:35,859 --> 00:43:44,478
show that one half of A+I is a...
630
00:43:45,306 --> 00:43:46,433
this is proof
631
00:43:51,591 --> 00:43:55,185
1/2 of A+I is a projection matrix
632
00:44:02,246 --> 00:44:03,126
Ok
633
00:44:07,031 --> 00:44:08,866
now let's see, what do I do?
634
00:44:11,888 --> 00:44:14,351
I could seek 2 ways to do this
635
00:44:14,351 --> 00:44:18,519
I could check the properties of the
projection matrix which are what?
636
00:44:18,554 --> 00:44:21,385
a projection matrix is symmetric
637
00:44:21,420 --> 00:44:24,776
Well that's certainly symmetric
because A is
638
00:44:24,811 --> 00:44:26,683
and what's the other property?
639
00:44:27,473 --> 00:44:30,977
I should square it and hopefully
get the same thing back
640
00:44:31,012 --> 00:44:33,210
so can I do that square and C
641
00:44:33,245 --> 00:44:34,923
I get the same thing back
642
00:44:35,532 --> 00:44:42,869
so If I squared
I get (see the board)
643
00:44:42,904 --> 00:44:51,331
right? and the question is
dose that agree with the thing itself
644
00:44:51,332 --> 00:44:53,241
1/2(A+ I)
645
00:45:00,027 --> 00:45:04,346
I guess, I like to know
something about A^2
646
00:45:05,128 --> 00:45:07,645
what is the A^2?
that is our problem
647
00:45:08,491 --> 00:45:09,970
what is A^2?
648
00:45:14,595 --> 00:45:16,835
if A is symmetric and orthogonal
649
00:45:16,870 --> 00:45:21,242
A is symmetric and orthogonal
650
00:45:25,236 --> 00:45:26,884
this is what we are given, right?
651
00:45:26,919 --> 00:45:29,693
it's symmetric, and also orthogonal
652
00:45:30,749 --> 00:45:32,634
so what is A^2?
653
00:45:34,330 --> 00:45:37,102
I, A^2 is I
654
00:45:37,915 --> 00:45:42,675
because A*A...
A equals to its own Inv[A]
655
00:45:42,710 --> 00:45:47,279
so AA=A*Inv[A]
656
00:45:47,314 --> 00:45:49,355
which is I
657
00:45:49,390 --> 00:45:52,259
so this A^2 here is I
658
00:45:55,612 --> 00:45:57,272
and now we've got it
659
00:45:57,998 --> 00:46:00,142
we've got 2 identities over 4
660
00:46:00,177 --> 00:46:01,248
that's good
661
00:46:01,283 --> 00:46:05,155
and we've got 2 As over 4
that's good, ok
662
00:46:05,190 --> 00:46:09,242
so it turned out to be
a projection matrix safely
663
00:46:09,277 --> 00:46:11,337
and we could also have said...
664
00:46:11,372 --> 00:46:13,869
Well, what are the eigenvalues
of this thing?
665
00:46:14,757 --> 00:46:18,231
what are the eigenvalues of (A+I)/2?
666
00:46:18,266 --> 00:46:21,230
if the eigenvalues of A are 1 and -1
667
00:46:21,265 --> 00:46:24,033
what are the eigenvalues of A+I?
668
00:46:25,377 --> 00:46:28,551
say with this in the last thirty
seconds here
669
00:46:29,415 --> 00:46:32,497
what are the...
if I know these eigenvalues of A
670
00:46:32,532 --> 00:46:38,716
and I add the identity
the eigenvalues of (A+I) are 0 and 2
671
00:46:38,751 --> 00:46:40,797
and then when I divide by 2
672
00:46:40,832 --> 00:46:43,428
the eigenvalues are 0 and 1
673
00:46:43,463 --> 00:46:46,417
so it is symmetric
and it's got the right eigenvalues
674
00:46:46,452 --> 00:46:48,303
it's a projection matrix
675
00:46:48,338 --> 00:46:52,636
Ok, you're seeing a lot of
stuff about eigenvalues
676
00:46:52,671 --> 00:46:55,084
and special matrices
677
00:46:55,119 --> 00:46:57,144
and that's what the quiz is about
678
00:46:57,964 --> 00:46:59,883
Ok, so good luck on the quiz
Last Modified 5/8/08 1:01 PM
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