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...One and the lecture on
symmetric matrices
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so that's the most important
class on matrices
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symmetric matrices,
A=Tran(A)
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so the first point, the main point of
the lecture I'll tell you right away
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what's special about the eigenvalues?
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what's special about the eigenvectors?
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this is the way we now
look at a matrix
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we want to know about these
eigenvalues and eigenvectors
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and if we have a
special type of a matrix
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that should tell us something about
the eigenvalues and eigenvectors
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like markup matrices,
they have an eigenvalue equal 1
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now, symmetric matrices
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can I just tell you right
of what the main fact
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the two main facts are the
eigenvalues of a symmetric matrix...
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Real, it's a real symmetric matrix
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talking mostly about the real matrices
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the eigenvalues are also real
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so our examples of rotation
matrices were...
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where we got the eigenvalues
were complex
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that won't happen now
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for symmetrices,
the eigenvalues are real
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and the eigenvectors are
also very special
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the eigenvectors are perpendicular
or orthogonal
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which do you prefer?
I'll say perpendicular
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Per...well, they are both long words
perpendi...
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Ok, right, so,
you should say why?
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and I'll at least answer
why for case 1
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maybe case 2, they're checking the...
the eigenvectors are perpendicular
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I'll leave to ....the book
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But let's just realize what...
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Well, first I have to say
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It could have a line for
the identity matrix
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there is a symmetric matrix
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Its eigenvalues are
certainly all real
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they are all 1 for the identity matrix
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what about the eigenvectors?
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Well, for the identity
every vector is an eigenvector
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so how can I say
they are perpendicular?
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what I really mean
is this word "are"
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should really be read
"can be chosen" perpendicular
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that is ...if we have...
it's the usual case
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if the eigenvalues are all different
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then each eigenvalue
as one line of eigenvectors
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that those lines are
perpendicular here
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but if an eigenvalue is repeated
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then it's the whole plane
of the eigenvectors
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and, oh, I'm saying is in that plane
we can choose perpendicular one
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so that's why it "can be chosen"
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is this in the case of
a repeated eigenvalue
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where there're some real
substantial freedoms
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but the typical case is different
eigenvalues are all real
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one dimentional eigenvector spaces,
eigenspaces are all perpendicular
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so let's just see the conclusion
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if we expect those as correct
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what happens, and I also mean that
there's a full set of them
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so that's part of this picture here
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this is a complete set
of eigenvectors
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perpendicular one
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so having a complete set of
eigenvectors means so normal...
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The usual case is that the matrix A
we can write in the terms of
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the eigenvalue matrix and its
eigenvector matrix this way, right?
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we can do that in the usual case
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but now what's special
when the matrix is symmetric
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so this is the usual case
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and now let me go to
the symmetric case
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so the symmetric case A
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this should become
somehow a little special
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Well, the λ's on the diangonal
is still on the diangonal
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they are real
but that's where they are
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what about the eigenvector matrix?
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so what can I do now special
about the eigenvector matrix?
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when the A itself is symmetric
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that says something good
about the eigenvector matrix
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so what does this lead to?
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these perpendicular eigenvectors
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I can only guarantee
they are perpendicular
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I could also make them unit vectors
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no problem, just scale
their length the one
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so what do I have?
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I have orthonomal eigenvectors
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and what does that tell me
about the eigenvector matrix?
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what letter should I now use
in place of X?
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member X has the eigenvectors
in its columns
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but now those columns
are orthonomal
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so the right letter to use is Q
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so that's where...
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so we've got the letter all set up
so this should be Q*λ*Inv(Q)
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Q standing in our mind always for
this matrix and in the case it's square
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It's...
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So these are the columns of Q
of course
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And, one more thing
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what's Inv(Q)?
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for a matrix who has
these orthonomal columns
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we know the inverse
is the same as the transpose
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so here is the beautiful...
there is the great description
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the factorization of a symmetric matrix
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and this is like one of the famous
theorems of the Linear Algebra
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that if I have a symmetric matrix
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it can be factored in this form
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an orthogonal matrix times diangonal
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times the transpose of
that orthogonal matrix
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and of course, everybody
immediately says
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yes, and if it is possible, then
that's clearly symmetric, right?
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we look at the products
of this three guys like that
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and taking their transpose
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and we've got it back again
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so you see the beauty
of this factorization
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it completely displays the eigenvalues
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and eigenvectors and the
symmetry of the whole thing
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because that product Q*λ*Tran(Q)
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if I transpose it
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this comes in this position
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and we get that matrix back again
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so that's in methematics
that's called spectrum theorem
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spectrum means the set
of eigenvalues of a matrix
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It somehow comes from the ideas of
lines of a combination of pure things
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where our matrix is broken down and
the pure eigenvalues and eigenvectors
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in mechanics it's often
called the principle axis theorem
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it's very useful
it means that if you have....
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we'll see it geometrically
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it means that if I have some material
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if I look at the right axis
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it become diangonals...
it becomes...
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the directions don't couple together
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Ok, so that's what we remember
from this lecture
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now I would like to say
why are the eigenvalues real?
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can I do that, so something
useful comes out
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so just come back,
come to that question
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Why real eigenvalues?
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Ok, so I have to start from
the only thing we know: Ax=λ*x
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Ok, but as far as I know
at this moment
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λ could be complex
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and I'm going to prove it's not
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and x could be complex
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and in fact, for the moment
even A could be ...
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we can even think, oh,
what happens if A is complex
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Well, one thing we can always do
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this is like always, always ok
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I can...If I have an equation, I can take
the complex conjugate of everything
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(see the board)
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I just mean that, everywhere
over here that there was an I
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and then I change it
to the minus I
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that you know the conjugate
business is (see the board)
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that's the meaning of conjugate
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and products behave right
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I can conjugate every factor
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so I haven't done everyting
yet except to say
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what would be true?
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if...
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in any case
even x or λ were complex
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of course, we're speaking about
real matrix A
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so I can take that out
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actually, this already tells me
something about real matrices
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I haven't use any assumption of
A=Tran(A) yet
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symmetry is waiting
in the willing to be used
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this tells me that if a real matrix has
an eigenvalue λ and and eigenvector X
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it also has another eigenvalue
it is λ bar with eigenvector x bar
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real matrices, the eigenvalues
come in λ, λ bar
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the complex eigenvalues come in λ
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and λ bar pairs
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but of course, I may mean to show
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they're not complex at all here
by getting symmetry in
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so how am I going to use symmetry
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I am going to transpose the equation to
(see the board)
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that is a number so I don't mind
where I put that number
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this is, then again, ok
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but now, I'm ready to use symmetry
I am ready...
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so this is all just mechanics
now comes the moment to say ok
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if a matrix is symmetric
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then this Tran(A) is the same as A
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you see at that moment
I use the assumption
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now let me finish the discussion
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here's where I finish
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I look at this original equation
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and I take the inner product
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I multiply both sides by...
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Oh, maybe I'll do it with this one
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00:12:51,910 --> 00:12:57,881
I multiply both sides by Tran(x bar)
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(see the board)
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00:13:05,986 --> 00:13:07,621
ok, fine
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Er...right?
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00:13:13,769 --> 00:13:15,467
now, what's the other one?
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00:13:15,502 --> 00:13:18,410
Oh, the other one I propably
use this guy
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Am I happy about this?
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00:13:23,650 --> 00:13:25,808
No, for some reason I'm not
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00:13:26,914 --> 00:13:33,596
I wonder if I take the inner product
of this from the right with the X bar
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00:13:33,631 --> 00:13:43,084
I get
(see the board)
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I've done something done,
because I've got...
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I haven't learned anything.
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00:13:54,820 --> 00:13:58,031
I've got...
those two equations are identical
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00:13:58,066 --> 00:14:01,302
Oh, forgive me for doing such a thing
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00:14:02,081 --> 00:14:04,913
but... I'll look at the book
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00:14:07,283 --> 00:14:11,602
Ok, so I took the dot product
194
00:14:11,637 --> 00:14:14,672
yeah, somehow
it's in...
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00:14:14,672 --> 00:14:17,449
I should have taken
a dot product of this guy here
196
00:14:17,484 --> 00:14:19,436
that's what I'm going to do
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00:14:19,471 --> 00:14:26,588
(see the board)
198
00:14:26,623 --> 00:14:31,699
Ok, that's fine
that comes directly from that
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00:14:31,734 --> 00:14:34,345
multiply both side by Tran(x bar)
200
00:14:34,380 --> 00:14:39,382
but now I'd like to get
why do I have x bar over there
201
00:14:41,272 --> 00:14:45,225
Ah, yes, forget this
202
00:14:46,092 --> 00:14:49,986
Ok, on this one, right, on this one
203
00:14:50,021 --> 00:14:53,287
I took it like that
I multiply on the right by x
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00:14:53,288 --> 00:14:55,357
That, the ideal
205
00:14:56,937 --> 00:14:57,812
Ok
206
00:14:59,745 --> 00:15:04,956
why am I happier
with this situation now?
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00:15:04,991 --> 00:15:07,011
a proof is coming here
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00:15:07,046 --> 00:15:12,615
because I compare
this guy with this one
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00:15:13,595 --> 00:15:15,636
and they have the same left-hand side
210
00:15:15,671 --> 00:15:18,131
so they have the same right-hand side
211
00:15:18,166 --> 00:15:19,632
so comparing those two
212
00:15:19,667 --> 00:15:22,559
I'll raise the board
to do this comparison
213
00:15:22,594 --> 00:15:39,070
(see the board)
214
00:15:39,105 --> 00:15:41,830
and the conclusion
I'm going to reach
215
00:15:41,865 --> 00:15:45,401
Am I on the right track here?
216
00:15:46,777 --> 00:15:50,808
the conclusion I'm going to reach is
λ=λ bar
217
00:15:55,470 --> 00:16:00,330
I will be happy to track down
the other possibly that this thing is 0
218
00:16:00,365 --> 00:16:02,662
but let me....oh
219
00:16:04,100 --> 00:16:06,393
oh, yeah, that's a part
that's not zero
220
00:16:07,197 --> 00:16:11,107
so once I know this is zero
221
00:16:11,142 --> 00:16:16,191
I just cancel it and I learned
λ=λ bar until what?
222
00:16:16,226 --> 00:16:22,462
Can you...have you got
reasoning all together?
223
00:16:24,307 --> 00:16:26,067
what does this tell us?
224
00:16:27,271 --> 00:16:30,721
λ is the eigenvalue of
this symmetric matrix?
225
00:16:31,607 --> 00:16:34,128
we just prove that it equals λ bar
226
00:16:34,163 --> 00:16:38,990
so we have just proved
that λ is real, right?
227
00:16:39,785 --> 00:16:44,029
if a number is equal
to its own complex conjugate
228
00:16:44,064 --> 00:16:46,300
then, there is no
imaginary part at all
229
00:16:46,335 --> 00:16:49,751
the number is real,
so λ is real
230
00:16:51,892 --> 00:16:54,606
Good, good
231
00:16:55,331 --> 00:17:00,015
Now, what...
but it depends on this expression
232
00:17:00,050 --> 00:17:05,026
I'm knowing that was
non-zero, so that I can cancel it out
233
00:17:05,061 --> 00:17:08,904
so can we just take a
second on that one?
234
00:17:08,939 --> 00:17:15,029
so this is an important quantity
x bar Tran(x)
235
00:17:15,064 --> 00:17:16,523
ok, now, remember
236
00:17:17,455 --> 00:17:21,122
as far as we know,
x is complex
237
00:17:22,603 --> 00:17:31,447
so this is, x is complex, has its
components x1,x2...xn
238
00:17:33,198 --> 00:17:36,776
and Tran[x bar]...
Well, it's transposed
239
00:17:38,829 --> 00:17:40,059
and it's conjugated
240
00:17:40,094 --> 00:17:47,225
(see the board)
241
00:17:47,260 --> 00:17:52,127
I'm really reminding you of
the crucial fact of complex numbers
242
00:17:52,162 --> 00:17:55,355
that will come into the next
lecture as well as this one
243
00:17:55,390 --> 00:18:01,319
so what can you tell me
about that product?
244
00:18:03,344 --> 00:18:07,452
I guess what I'm going to say
is if I have a complex vector
245
00:18:08,295 --> 00:18:11,535
this would be the quantity
I would like
246
00:18:11,536 --> 00:18:13,615
This is the quantity I'd like
247
00:18:13,650 --> 00:18:17,279
I'll take this vector,
times its transpose....
248
00:18:17,314 --> 00:18:22,042
Now what happens usually
if I take a vector, x Tran(x)
249
00:18:22,907 --> 00:18:26,110
and that is a quatity
we see all the time, x Tran(x)
250
00:18:26,145 --> 00:18:28,936
that's the link of x^2, right?
251
00:18:29,211 --> 00:18:31,702
that's positive link square
252
00:18:31,702 --> 00:18:35,773
it's x1 square,
process 2 square, and so on
253
00:18:35,773 --> 00:18:39,032
now, our vectors are complex
254
00:18:39,067 --> 00:18:43,767
you see the effect, I am conjugating
one of these guys
255
00:18:43,802 --> 00:18:45,545
So now, when I do the multiplication
256
00:18:45,580 --> 00:19:02,428
(see the board)
257
00:19:03,059 --> 00:19:05,782
I mean,
what's the point?
258
00:19:06,742 --> 00:19:08,320
what's the point?
259
00:19:08,320 --> 00:19:10,908
when I multiply a number
by its conjugate
260
00:19:10,943 --> 00:19:14,145
a complex number by its conjugate,
what do I get?
261
00:19:15,668 --> 00:19:20,555
I get...the imaginary part is gone.
262
00:19:20,590 --> 00:19:24,016
when I multiply (a+ib)
by its conjugate
263
00:19:24,808 --> 00:19:29,247
what's the result of that of each
of those separated multiplications
264
00:19:29,571 --> 00:19:31,276
there is a a^2
265
00:19:32,738 --> 00:19:38,212
and what's b^2...
comes in with a plus or a minus?
266
00:19:38,482 --> 00:19:39,637
a plus
267
00:19:40,355 --> 00:19:43,831
(see the board)
268
00:19:43,866 --> 00:19:46,097
And what about the imaginary part?
269
00:19:48,514 --> 00:19:49,680
Gone, right?
270
00:19:50,614 --> 00:19:53,856
And iab and -iab
271
00:19:53,891 --> 00:19:57,512
so this is the right thing to do
272
00:19:57,547 --> 00:19:59,561
if you want a decent answer
273
00:20:02,601 --> 00:20:06,940
then multiply numbers
by their conjugates
274
00:20:07,191 --> 00:20:12,990
multiply vectors by
their conjugates of Tran(x)
275
00:20:13,677 --> 00:20:18,505
so this quantity is positive,
this quantity is positive
276
00:20:18,540 --> 00:20:21,926
the whole thing is positive
except the zero vector
277
00:20:22,730 --> 00:20:27,435
and that allows me to know
that this is a positive number
278
00:20:28,344 --> 00:20:31,816
which I safely cancel out
then I reach the conclusion
279
00:20:31,851 --> 00:20:36,061
so actually in this discussion here
280
00:20:36,522 --> 00:20:38,217
I've got two things
281
00:20:38,252 --> 00:20:41,364
I reach the conclusion that λ is real
282
00:20:41,399 --> 00:20:42,503
which I wanted to do
283
00:20:43,249 --> 00:20:44,808
but at the same time
284
00:20:44,843 --> 00:20:48,912
we shall solve what to do
if the things were complex
285
00:20:49,097 --> 00:20:59,546
if the vector is complex, then its
Tran(x bar), this is length squared
286
00:21:00,164 --> 00:21:08,118
and as I said the next lecture Monday
will reapeat this right thing
287
00:21:08,153 --> 00:21:10,158
and then do the right thing
for matrices
288
00:21:10,193 --> 00:21:14,566
and all other complex possibilities
289
00:21:15,515 --> 00:21:23,900
Ok, but the main point then is that
the eigenvalues of a symmetric matrix
290
00:21:24,733 --> 00:21:28,279
just...where did we use
symmetry, by the way?
291
00:21:28,834 --> 00:21:30,205
we use it here, right?
292
00:21:31,060 --> 00:21:32,850
can I just...
293
00:21:34,980 --> 00:21:36,642
suppose A was a complex
294
00:21:37,251 --> 00:21:39,631
suppose A has been a complex number
295
00:21:39,666 --> 00:21:42,342
could I make all these work?
296
00:21:42,377 --> 00:21:48,914
if A is was complex number, complex matrix
then here I shouldn't write A bar
297
00:21:50,020 --> 00:21:53,070
I raise the bar
because A was real
298
00:21:53,105 --> 00:21:55,229
but now let's suppose
for a moment it's not
299
00:21:55,774 --> 00:21:57,702
then when I take this step
300
00:21:57,994 --> 00:21:59,760
what should I have
301
00:22:00,636 --> 00:22:03,090
what did I do on that step
by transpose?
302
00:22:03,125 --> 00:22:06,273
so I should have Tran(A bar)
303
00:22:08,863 --> 00:22:10,960
in the symmetric case,
that was A
304
00:22:11,641 --> 00:22:13,875
and that's what made
everything work, right?
305
00:22:13,910 --> 00:22:16,522
this lead me easy to that
306
00:22:17,667 --> 00:22:21,073
this one
let me lead to this
307
00:22:21,380 --> 00:22:23,217
when the matrix was real
308
00:22:23,252 --> 00:22:24,627
so that didn't matter
309
00:22:24,662 --> 00:22:26,908
and it's symmetric,
so that didn't matter
310
00:22:27,358 --> 00:22:28,376
then I've got A
311
00:22:28,954 --> 00:22:31,745
but...
so now I just get to ask you
312
00:22:32,009 --> 00:22:35,301
suppose the matrix
had been complex
313
00:22:36,053 --> 00:22:41,536
what is the right
equivalent symmetry?
314
00:22:41,571 --> 00:22:48,029
so here...let me say
good matrices
315
00:22:50,649 --> 00:22:53,896
by good... I mean real λ's
316
00:22:55,636 --> 00:23:00,043
and perpendicular x's
317
00:23:02,514 --> 00:23:06,539
and tell me now which matrices are good
318
00:23:08,844 --> 00:23:10,266
If they are symme...
If they are...
319
00:23:11,075 --> 00:23:12,839
if they're real matrices
320
00:23:12,874 --> 00:23:14,706
the good ones are symmetric
321
00:23:14,741 --> 00:23:16,186
because when everything run through
322
00:23:16,221 --> 00:23:17,835
so the good...
323
00:23:17,870 --> 00:23:20,320
I'm saying now what's good
324
00:23:20,642 --> 00:23:23,239
This...these are good matrices
325
00:23:23,274 --> 00:23:26,314
they have real eigenvalues
perpendicualr eigenvectors
326
00:23:26,349 --> 00:23:32,569
good means A=Tran(A)
if real
327
00:23:33,383 --> 00:23:36,084
then what is our proof work?
328
00:23:36,291 --> 00:23:38,103
but if A is complex
329
00:23:38,875 --> 00:23:43,933
Our proof will still work
provided Tran(A bar) is A
330
00:23:44,673 --> 00:23:46,362
you see that I'm saying
331
00:23:47,204 --> 00:23:50,338
I'm saying if we have
complex matrices
332
00:23:51,895 --> 00:23:56,648
and we want to say are they
as good as symmetric matrices?
333
00:23:57,545 --> 00:24:04,100
then, we should not only transpose
the thing but conjugate it
334
00:24:04,135 --> 00:24:06,368
those are good matrices
335
00:24:07,248 --> 00:24:11,358
and of course, the most important
case is
336
00:24:11,393 --> 00:24:14,703
when they are real,
this part doesn't matter
337
00:24:14,738 --> 00:24:17,496
and I just have A and
Tran(A) symmetric
338
00:24:17,531 --> 00:24:20,707
I'll just repeat that
339
00:24:21,441 --> 00:24:25,284
the good matrices,
if complex, are these
340
00:24:26,225 --> 00:24:29,827
if real, that doesn't
make any difference
341
00:24:29,862 --> 00:24:31,821
so I'm just saying symmetric
342
00:24:31,856 --> 00:24:38,691
and of course, 99% of the examples
and applications, the matrices are real
343
00:24:38,726 --> 00:24:43,533
and we don't have that, and then
symmetric is the key property
344
00:24:44,240 --> 00:24:50,990
Ok, so, that's these main facts
345
00:24:51,920 --> 00:24:57,938
Now let me just...
so that this (X bar)Tran(X)
346
00:24:58,473 --> 00:25:05,150
Ok, so I just write it
once more in this form
347
00:25:05,185 --> 00:25:09,362
so perpendicular, orthnomal eigenvectors
348
00:25:09,397 --> 00:25:14,380
real eigenvalues, transposes
are orthnomal eigenvectors
349
00:25:15,278 --> 00:25:17,407
that's the symmetric case
350
00:25:17,442 --> 00:25:19,095
A=Tran(A)
351
00:25:19,869 --> 00:25:22,441
Ok, good
352
00:25:24,443 --> 00:25:28,477
actually, I even take one
more step here
353
00:25:29,288 --> 00:25:31,248
suppose...
354
00:25:31,812 --> 00:25:35,909
I can break this down to
show you really
355
00:25:35,944 --> 00:25:38,425
what's that says about
the symmetric matrix
356
00:25:40,727 --> 00:25:42,421
I can break them down,
let me...
357
00:25:42,456 --> 00:25:46,027
here goes the eigenvectors
358
00:25:48,546 --> 00:25:53,799
here goes these eigenvalues
λ1, λ2, and so on...
359
00:25:53,834 --> 00:25:56,224
here goes these eigenvectors
360
00:25:56,755 --> 00:25:58,061
transposed...
361
00:26:01,528 --> 00:26:06,355
And what happens if I
actually do out that multiplication?
362
00:26:08,035 --> 00:26:09,556
you see what will happpen...
363
00:26:09,920 --> 00:26:13,744
there's λ1 times Tran(q1)
364
00:26:14,209 --> 00:26:18,198
so the first row here is
just λ1Tran(q1)
365
00:26:18,577 --> 00:26:21,617
if I multiplies columns times row
366
00:26:21,871 --> 00:26:23,600
remember I could do that
367
00:26:24,287 --> 00:26:30,292
when I multiply matrices, I can
multiply columns times rows
368
00:26:30,557 --> 00:26:36,916
so what I do that, I get λ1
and then the column, and then the row
369
00:26:37,729 --> 00:26:41,821
and then λ2, the column and the row
370
00:26:48,470 --> 00:26:52,443
every symmetric matrix
breaks up into these pieces
371
00:26:53,598 --> 00:26:56,804
so these pieces have real λs
372
00:26:57,412 --> 00:27:04,330
and they have these
orthonomal eigenvectors
373
00:27:07,353 --> 00:27:12,530
and maybe you even could tell me
what kind of matrix have I got there
374
00:27:14,166 --> 00:27:16,311
suppose I take a unit vector
375
00:27:17,093 --> 00:27:19,304
times its transpose
376
00:27:19,339 --> 00:27:22,348
so columns times row,
I'm getting a matrix
377
00:27:24,199 --> 00:27:27,253
and that matrix has
a special name
378
00:27:28,997 --> 00:27:34,174
what kind of matrix is it, we've seen
those matrices now in Chapter 4
379
00:27:34,896 --> 00:27:42,345
it is ATran(A) with a unit vector
so I don't have to divide by ATran[A]
380
00:27:42,982 --> 00:27:47,368
that matrix is a projection matrix
381
00:27:47,403 --> 00:27:49,921
that's a projection matrix
it's symmetric
382
00:27:50,663 --> 00:27:52,120
and if I square it
383
00:27:52,155 --> 00:27:56,535
to be a q1Tran(q1) which is 1
384
00:27:57,211 --> 00:27:59,485
so I'll get that matrix back again
385
00:28:00,369 --> 00:28:14,828
so every symmetric matrix is a
combination of...
386
00:28:15,512 --> 00:28:28,217
of mutually perpendicular
projection matrices, ok
387
00:28:29,060 --> 00:28:34,837
that's another way that people like
to think of the spectrum theorem
388
00:28:34,872 --> 00:28:39,045
that every symmetric matrix
can be broke down in that way
389
00:28:40,294 --> 00:28:41,724
I guess at this moment
390
00:28:42,469 --> 00:28:43,994
first I haven't done an example
391
00:28:44,029 --> 00:28:46,619
I could create a symmetric matrix
392
00:28:47,397 --> 00:28:51,466
check and find its eigenvalues,
they will come out real
393
00:28:51,501 --> 00:28:54,815
find the eigenvectors,
they will come out perpendicular
394
00:28:54,850 --> 00:28:57,155
and you will see it in numbers
395
00:28:57,190 --> 00:29:00,113
but maybe I'll leave it here
for the moment in latters
396
00:29:01,130 --> 00:29:05,773
Oh, maybe I will do it
with numbers for this reason
397
00:29:05,808 --> 00:29:08,345
because of one more remarkable fact
398
00:29:09,703 --> 00:29:15,307
can I just put this further great fact
about symmetric matrices on the board
399
00:29:18,667 --> 00:29:22,758
when I have symmetric matrices
I know their eigenvalues are real
400
00:29:23,655 --> 00:29:26,494
so then I can get interested
in the question
401
00:29:26,529 --> 00:29:28,239
are they positive or negative?
402
00:29:29,060 --> 00:29:31,411
and you rememer why that's important
403
00:29:31,446 --> 00:29:36,167
for differential equation that decide
between instability and stability
404
00:29:37,124 --> 00:29:41,626
after I know they are real
then the next question is
405
00:29:41,661 --> 00:29:43,429
are they positive or are they negative
406
00:29:44,960 --> 00:29:52,374
and I hate to have to compute those
eigenvalues to answer that question
407
00:29:52,933 --> 00:29:53,704
right?
408
00:29:53,739 --> 00:29:58,751
because computing the eigenvalues of
a symmetric matrix of an order say fifty
409
00:29:58,786 --> 00:30:04,827
compute fifty eigenvalues is a job
410
00:30:05,803 --> 00:30:11,165
I mean by penciling paper...
it's a lifetime job
411
00:30:11,200 --> 00:30:14,289
I mean, which...
and in fact...
412
00:30:16,782 --> 00:30:20,111
a few years ago, well say,
20 years ago
413
00:30:20,752 --> 00:30:24,131
or thirty, nobody really
knew how to do it
414
00:30:24,690 --> 00:30:29,150
I mean so like science was
stuck on this problem
415
00:30:29,550 --> 00:30:32,321
if you have a matrix
of order 50 or 100
416
00:30:32,356 --> 00:30:34,178
how do you find these eigenvalues?
417
00:30:35,500 --> 00:30:37,057
numerically now, I'm just saying
418
00:30:37,092 --> 00:30:38,530
Because penciling papers...
419
00:30:38,565 --> 00:30:44,131
we will run out of time or paper
or something before we get it
420
00:30:46,724 --> 00:30:49,263
Well, and you might think, ok
421
00:30:49,298 --> 00:30:57,255
get MATLAB to compute
the determinant of λ-A, A-λI
422
00:30:57,290 --> 00:30:59,996
this polynomial of 50th degree
423
00:31:00,031 --> 00:31:01,506
and then find the roots
424
00:31:03,124 --> 00:31:06,688
MATLAB will do it,
but it will complain
425
00:31:06,921 --> 00:31:11,772
because it is a very bad
way to find the eigenvalues
426
00:31:11,807 --> 00:31:14,558
I'm sorry to be saying this
427
00:31:14,757 --> 00:31:17,654
because it is the way
I taught you to do it, right?
428
00:31:17,689 --> 00:31:19,685
I taught you to find the eigenvalues
429
00:31:19,720 --> 00:31:24,000
by doing that determinant and
taking the roots of that polynomial
430
00:31:24,035 --> 00:31:29,099
but now I'm saying ok
I'm really meant that for 2*2, 3*3...
431
00:31:29,134 --> 00:31:32,463
but I didn't mean you
do it on 50*50
432
00:31:32,498 --> 00:31:37,311
and you're not too unhappy
because you didn't want to do it
433
00:31:37,346 --> 00:31:44,293
but, good, because it would
be a very unstable way
434
00:31:44,328 --> 00:31:47,887
the fifty answers that'll come out
will be highly unreliable
435
00:31:47,922 --> 00:31:54,147
so new ways are much better
than find the fifty eigenvalues
436
00:31:54,182 --> 00:31:56,984
that's a part of numerical
linear algebra
437
00:31:57,685 --> 00:32:00,308
but here is the remarkable fact
438
00:32:02,098 --> 00:32:06,171
that MATLAB was quite happy
to find the fifty pivots
439
00:32:07,000 --> 00:32:10,811
right? now the pivots are not
the same as eigenvalues
440
00:32:12,562 --> 00:32:13,957
but here is the great thing
441
00:32:13,992 --> 00:32:15,873
if I had a real matrix
442
00:32:16,654 --> 00:32:18,610
I could find those fifty pivots
443
00:32:18,645 --> 00:32:22,111
and I could see maybe 28
of them are positive
444
00:32:22,146 --> 00:32:24,106
and 22 are negative pivots
445
00:32:24,141 --> 00:32:27,282
and I can compute those
safely and quickly
446
00:32:27,647 --> 00:32:32,389
and the great fact is the 28 of
the eigenvalues will be positive
447
00:32:32,424 --> 00:32:34,577
and 22 will be negative
448
00:32:35,579 --> 00:32:37,046
that's the signs of the pivots
449
00:32:37,081 --> 00:32:38,319
so this is like a...
450
00:32:38,636 --> 00:32:42,441
I hope you think
that is just a kind of nice thing
451
00:32:42,476 --> 00:32:48,861
that the signs of the
pivots for symmetric
452
00:32:48,896 --> 00:32:51,270
I'm always talking
about symmetric matrices
453
00:32:51,961 --> 00:32:53,995
I really like, trying to convince
you that
454
00:32:54,030 --> 00:32:57,727
the symmetric matrices
are better than the rest
455
00:32:57,762 --> 00:33:09,468
so the signs of the pivots are same
as the signs of the eigenvalues
456
00:33:10,311 --> 00:33:11,759
the same number
457
00:33:12,693 --> 00:33:17,380
the number of pivots
greater than 0
458
00:33:19,103 --> 00:33:28,795
the number of possible pivots is equal
to the number of possible eigenvalues
459
00:33:29,551 --> 00:33:32,795
so that actually is a very useful...
460
00:33:32,830 --> 00:33:38,870
that gives you a good start on a decent
way to compute eigenvalues
461
00:33:39,632 --> 00:33:41,568
because you can narrow them down
462
00:33:41,603 --> 00:33:43,450
you can find out
how many are positive
463
00:33:43,485 --> 00:33:44,564
how many are negative
464
00:33:45,142 --> 00:33:49,564
then you could shift the matrix
by 7 times the identity
465
00:33:50,558 --> 00:33:53,098
that will shift all the eigenvalues
by 7
466
00:33:54,011 --> 00:33:56,470
then you could take the
pivots of that matrix
467
00:33:56,505 --> 00:34:02,721
then you would know how many eigenvalues
of the original were above 7 or below 7
468
00:34:02,964 --> 00:34:06,114
so this neat little theorem
469
00:34:06,149 --> 00:34:13,723
that the symmetric matrices
have this connection between ...
470
00:34:13,758 --> 00:34:15,068
nobody is mixing up
471
00:34:15,103 --> 00:34:17,418
thinking the pivots are the eigenvalues
472
00:34:18,530 --> 00:34:23,455
the only thing I can think of
is the product of the pivots
473
00:34:23,490 --> 00:34:25,797
equal to the product
of the eigenvalues
474
00:34:25,832 --> 00:34:26,722
why is that?
475
00:34:27,578 --> 00:34:29,406
So I ask you on the reason for that
476
00:34:29,441 --> 00:34:33,224
why is the product of the pivots
for a symmetric matrix
477
00:34:33,259 --> 00:34:35,146
is the same as the product
of the eigenvalues
478
00:34:36,415 --> 00:34:43,418
beacuse they both equal
the determinant, right?
479
00:34:43,453 --> 00:34:47,225
the product of the pivots
give determinant, if no row exchanges
480
00:34:47,260 --> 00:34:50,700
the product of the eigenvalues
always give the determinant
481
00:34:50,735 --> 00:34:53,974
so the products...
482
00:34:54,009 --> 00:34:57,599
but that doesn't tell you anything
about the fifty individual one
483
00:34:57,634 --> 00:34:59,261
which this does
484
00:34:59,296 --> 00:35:05,774
Ok, so those are central factors
about the symmetric matrices
485
00:35:06,552 --> 00:35:15,013
Ok, now I said in the lecture
description that
486
00:35:15,013 --> 00:35:23,063
I would take the last minutes
to start on positive definite matrices
487
00:35:23,098 --> 00:35:24,565
Because we are right there
488
00:35:24,600 --> 00:35:31,173
we are ready to say what is
the postive definite matrix
489
00:35:39,853 --> 00:35:42,114
It's symmetric, first of all.
490
00:35:43,013 --> 00:35:46,272
always I will mean symmetric
491
00:35:48,728 --> 00:35:52,274
so this is the next
section of the book
492
00:35:52,275 --> 00:35:53,745
It's about this...
493
00:35:53,745 --> 00:35:56,755
if symmetric matrices are good
494
00:35:57,732 --> 00:36:00,424
which is like the point
of my lecture so far
495
00:36:01,117 --> 00:36:11,714
then positive definite matrices
are subclass that are excellent, ok
496
00:36:11,749 --> 00:36:13,933
just the greatest
497
00:36:13,968 --> 00:36:19,722
So, what are...their matrices,
their symmetric matrices
498
00:36:19,757 --> 00:36:22,118
so all their eigenvalues are real
499
00:36:22,153 --> 00:36:23,897
you can guess what they are
500
00:36:23,932 --> 00:36:31,260
these are symmetric matrices
with all the eigenvalues are ...
501
00:36:33,930 --> 00:36:36,905
Ok, tell me what is the right
502
00:36:39,822 --> 00:36:44,742
what, well, it's hinted, of course
by the name for these things
503
00:36:44,777 --> 00:36:47,639
all the eigenvalues are positive
504
00:36:48,426 --> 00:36:49,500
ok
505
00:36:53,573 --> 00:36:55,432
tell me about the pivots
506
00:36:55,467 --> 00:36:58,761
we can check the eigenvalues
or we can check the pivots
507
00:36:58,796 --> 00:37:02,718
all the pivots are what?
508
00:37:07,281 --> 00:37:11,038
and then I'll finally give
an example I feel awful
509
00:37:11,073 --> 00:37:15,645
but I've got to this point in a lecture
and I haven't given you a single example
510
00:37:15,680 --> 00:37:17,009
so let me give you one
511
00:37:17,306 --> 00:37:21,762
(see the blackboard)
512
00:37:22,424 --> 00:37:25,813
an symmetric...
fine...
513
00:37:26,104 --> 00:37:29,922
its eigenvalues are real...
for sure
514
00:37:30,602 --> 00:37:32,403
but more than that
515
00:37:33,223 --> 00:37:36,000
I know the signs of
those eigenvalues
516
00:37:37,595 --> 00:37:41,401
and also I know the
signs of those pivots
517
00:37:41,436 --> 00:37:43,130
what's the deal with the pivots?
518
00:37:43,165 --> 00:37:46,161
the eigenvalues are all positive
519
00:37:46,196 --> 00:37:50,017
and if this little fact is true
520
00:37:50,052 --> 00:37:53,833
that the pivots and eigenvalues
are of the same sign
521
00:37:53,868 --> 00:37:57,984
then this must be true
all the pivots are positive
522
00:38:00,545 --> 00:38:02,959
and that's the good way to test
523
00:38:02,994 --> 00:38:05,048
this is a good test because I can...
524
00:38:05,083 --> 00:38:07,033
what are the pivots for that matrix?
525
00:38:08,060 --> 00:38:10,911
the pivots for that matrix are 5
526
00:38:11,581 --> 00:38:16,422
so pivots are 5...
527
00:38:16,457 --> 00:38:17,944
what is the second pivots?
528
00:38:17,979 --> 00:38:24,143
have we like noticed the formula
for the second pivot in the matrix?
529
00:38:27,679 --> 00:38:31,463
It doesn't necessary, you know,
make some fraction for sure
530
00:38:31,498 --> 00:38:33,307
but what is that fraction
531
00:38:33,342 --> 00:38:34,323
can you tell me?
532
00:38:34,358 --> 00:38:38,536
Well, here, the product of
the pivots is the determinant
533
00:38:39,361 --> 00:38:41,413
what is the determinant of this matrix?
534
00:38:44,000 --> 00:38:50,181
11? so the second
pivot must be 11/5
535
00:38:51,058 --> 00:38:52,768
So that the product is 11
536
00:38:53,486 --> 00:38:55,163
they're both positive
537
00:38:56,352 --> 00:39:00,572
then I know that the eigenvalues of
that matrix are both positive
538
00:39:00,607 --> 00:39:02,364
what are the eigenvalues?
539
00:39:02,399 --> 00:39:04,400
Well, I've got take the roots of ...
540
00:39:04,435 --> 00:39:39,476
(see the blackboard)
541
00:39:41,774 --> 00:39:43,792
the eigenvalues well, 2*2
542
00:39:43,827 --> 00:39:46,867
they're not so terrible
but they're not so perfect
543
00:39:46,902 --> 00:39:49,920
pivots are really simple
544
00:39:53,783 --> 00:40:00,178
and this is the family of matrices that you
really want in differential equations
545
00:40:00,630 --> 00:40:05,110
because you know the signs
of the eigenvalues
546
00:40:05,145 --> 00:40:06,807
so you know the stability or not
547
00:40:07,700 --> 00:40:13,109
Ok, there is one another related
fact I can pop in here
548
00:40:13,144 --> 00:40:17,609
in the time availiable
for positive definite matrices
549
00:40:19,501 --> 00:40:23,703
The related fact is to ask you
about the determinant
550
00:40:23,738 --> 00:40:25,969
so what's the determinant?
551
00:40:34,502 --> 00:40:36,213
what can you tell me if I...
552
00:40:36,248 --> 00:40:39,492
remember positive definite means
553
00:40:39,527 --> 00:40:42,952
all eigenvalues are positive
all pivots are positive
554
00:40:42,987 --> 00:40:45,218
so what can you tell me
about the determinant
555
00:40:45,848 --> 00:40:47,682
it's positive too
556
00:40:49,689 --> 00:40:53,578
but somehow,
that's not quite enough
557
00:40:54,344 --> 00:40:55,979
here is the matrix
558
00:40:56,014 --> 00:40:59,624
(see the blackboard)
559
00:41:00,740 --> 00:41:02,744
what is the determinant of that guy?
560
00:41:03,666 --> 00:41:05,238
it's positive, right?
561
00:41:05,273 --> 00:41:07,645
is this a positive definite matrix?
562
00:41:07,680 --> 00:41:09,545
the pivots,
what are the pivots?
563
00:41:09,580 --> 00:41:10,794
well, negative
564
00:41:11,573 --> 00:41:15,062
what are the eigenvalues?
Well, they are also the same
565
00:41:15,097 --> 00:41:22,611
so somehow, I don't just want the
determinant of the whole matrix
566
00:41:22,646 --> 00:41:24,178
here is 11,
that's great
567
00:41:24,213 --> 00:41:27,955
here the determinant of the
whole matrix is 3, that's positive
568
00:41:29,705 --> 00:41:35,552
I also...I've got to check
like little subdeterminant
569
00:41:36,697 --> 00:41:39,080
They maybe coming down from the left
570
00:41:39,115 --> 00:41:42,760
so the 1*1 and 2*2
have to be positive
571
00:41:42,795 --> 00:41:45,387
so there, that's where I get the all
572
00:41:46,323 --> 00:41:50,645
All, can I call them
subdeterminants?
573
00:41:50,646 --> 00:41:55,222
Do I need to make the thing proved
574
00:41:55,347 --> 00:42:00,776
I need a test and think
not just the big determinant.
575
00:42:00,811 --> 00:42:06,254
all subdeterminant are positive,
then I'm ok
576
00:42:08,370 --> 00:42:09,644
Then I'm ok
577
00:42:10,292 --> 00:42:12,011
this passes the test
578
00:42:12,995 --> 00:42:16,306
5 is positive,
and 11 is positive
579
00:42:16,341 --> 00:42:20,653
this fails the test because
that -1 there is negative
580
00:42:21,726 --> 00:42:24,619
and then the big determinant
is positive 3
581
00:42:25,743 --> 00:42:34,227
so this fact, you see that actually
the course like coming together
582
00:42:36,878 --> 00:42:38,795
and that's really my point now
583
00:42:38,830 --> 00:42:44,461
and next in this lecture, and
particularly next Wednesday or Friday
584
00:42:45,773 --> 00:42:51,231
the course comes together these pivots
that we met in the first week
585
00:42:52,131 --> 00:42:55,719
these determinants that we
met in the middle of the course
586
00:42:56,418 --> 00:43:00,281
these eigenvalues that
we met most recently
587
00:43:00,316 --> 00:43:02,934
all matrices are square here
588
00:43:03,776 --> 00:43:09,154
so coming together for square matrices
mean these three pieces come together
589
00:43:09,189 --> 00:43:14,358
and they come together
in that beautiful fact that
590
00:43:14,393 --> 00:43:18,443
if I had one of these,
I have the others
591
00:43:19,371 --> 00:43:22,131
but for symmetric matrices
592
00:43:22,166 --> 00:43:27,653
so this will be the positive
definite section
593
00:43:27,688 --> 00:43:31,931
and then the real climax
of the course is
594
00:43:31,966 --> 00:43:36,861
to make everything come
together for n*n matrices
595
00:43:37,819 --> 00:43:39,776
not necessarily symmetric
596
00:43:39,811 --> 00:43:44,900
bring everything together there,
and that will be the final thing
597
00:43:44,935 --> 00:43:50,883
Ok, so have a great weekend and
don't forget symmetric matrices.
598
00:43:50,918 --> 00:43:51,764
Thanks.
Last Modified 4/5/08 12:39 AM
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