GILBERT STRANG: OK.
So this video is about using
eigenvectors and eigenvalues
to take powers of
a matrix, and I'll
show you why we want to
take powers of a matrix.
And then the next
video would be using
eigenvalues and eigenvectors to
solve differential equations.
The two big applications.
So here's the first application.
Let me remember the main facts.
That if A-- if.
This is an important point.
Not every matrix has n
independent eigenvectors that
would go into matrix
V. You remember
V is the eigenvector
matrix, and I
need n independent eigenvectors
in order to have a V inverse,
to make that formula correct.
So that's the key
formula for using
eigenvalues and eigenvectors.
And the case where we might
run short of eigenvectors
is when maybe one
eigenvalue is repeated.
It's a double eigenvalue,
and maybe there's
only one eigenvector
to go with it.
Every eigenvalue's got at
least one line of eigenvectors.
But we might not have two when
the eigenvalue is repeated
or we might.
So there are cases when
this formula doesn't apply.
Because I must be able
to take V inverse,
I need n independent
columns there.
OK.
But when it works,
it really works.
So the n-th power,
just remembering,
is V lambda V inverse, V
lambda V inverse, n times.
But every time I have
a V inverse and a V,
that's the identity.
So I move V out
at the beginning.
I have lambda, lambda,
lambda, n of those,
and a V inverse at the very end.
So that's the nice result for
the n-th power of a matrix.
Now I have to show you
how to use that formula,
how to use eigenvalues
and eigenvectors.
OK.
So we know we can take
powers of a matrix.
So first of all, what
kind of equation?
There's an equation.
That's called a
difference equation.
It goes from step k to step
k plus 1 to step k plus 2.
It steps one at a time and
every time multiplies by A.
So after k steps, I've
multiplied by A k times
from the original u0.
So instead of a
differential equation,
it's a step difference
equation with u0 given.
And there's the solution.
That's the quickest
form of the solution.
A to the k-th power,
that's what we want.
But just writing
A to the k, if we
had a big matrix, to
take its hundredth power
would be ridiculous.
But with eigenvalues
and eigenvectors,
we have that formula.
OK.
But now I want to think.
Let me try to turn that
formula into something
that you just naturally see.
And we know what happens.
If u0 is an eigenvector,
if u0 is an eigenvector,
that probably won't happen
because there are just
n eigenvector directions.
But if it happened to be an
eigenvector, then every step
we'd multiply by lambda, and
we'd have the answer, lambda k
times.
But what do we do for all
the initial vectors u0 which
are maybe not an eigenvector?
How do I proceed?
How do I use eigenvectors when
my original starting vector
is not an eigenvector?
And the answer is, it will be
a combination of eigenvectors.
So making this formula
real starts with this.
So I write u0 as a combination
of the eigenvectors.
And I can do it
because if I have
n independent eigenvectors,
that will be a basis.
Every vector can be
written in the basis.
So I'm looking there at a
combination of eigenvectors.
And now the point is that as
I take these steps to u1--
what will u1 be?
u1 will be Au0.
So I'm multiplying by A. So
when I multiply this by A,
what happens?
That's the whole point.
c1, A times x1 is
lambda 1 times x1.
It's an eigenvector.
c2 tells me how much of the
second eigenvector I have.
When I multiply by A, that
multiplies by lambda 2,
and so on, cn lambda n xn.
And that's the thing.
Each eigenvector
goes its own way,
and I just add them together.
OK.
And what about A
to the k-th power?
Now, that will give me uk.
And what happens if
I do this k times?
You've seen what I got
after doing it one time.
If I do it k times,
that lambda 1
that multiplies its eigenvector
will happen k times.
So I'll have lambda
1 to the k-th power.
Do you see that?
Every step brings
another factor lambda 1.
Every step brings
another factor lambda 2.
Every step brings--
that's the answer.
That is-- well, that answer
must be the same as this answer.
And I'll do an
example in a minute.
Right now, I'm just getting
the formulas straight.
So I have the quickest
possible formula,
but it doesn't help me much.
I have the using the
eigenvectors and eigenvalue
formula.
And here I have it
that, really, it's
the same thing written out as
a combination of eigenvectors.
And then this is my answer.
That's my answer to the--
that's my solution uk.
That's it.
So that must be
the same as that.
Do you want to just
think for one minute
why this answer is the
same as that answer?
Well, we need to know
what are the c's?
Well, the c's came from u0.
And if I write that equation
for the c's-- do you see what I
have as an equation for the c's?
u0 is this combination
of eigenvectors.
That's a matrix multiplication.
That's the eigenvector matrix
multiplied by the vector
c of coefficients, right?
That's how a matrix
multiplies a vector.
The columns, which are the x's,
multiply the numbers c1, c2,
cn.
There it is.
That's the same as that.
So u0 is Vc.
So c is V inverse u0.
Oh, that's nice.
That's telling us what
are the coefficients, what
are the numbers, what
amount of each eigenvector
is present in u0.
This is the equation.
But look, you see
there that V inverse
u0, that's the first part
there of the formula.
I'm trying to match this
formula with that one.
And I'm taking one
step to recognize
that this part of the
formula is exactly c.
You might want to
think about that.
Run this video once more
just to see that step.
Now what do we do?
We've got the lambdas.
So I'm taking care of
the c's, you could say.
Now I need the lambda
to the k-th power--
lambda 1 to the k-th, lambda
2 to the k-th, lambda n
to the k-th.
That's exactly
what goes in here.
So that factor is producing
the lambdas to the k-th power.
And finally, this factor has--
everybody's remembering here.
V is the eigenvector
matrix x1, x2, to xn.
And when I multiply by V,
it's a matrix times a vector.
This is a matrix.
This is a vector.
And I get the combination--
I'm adding up.
I'm reconstructing the solution.
So first I break
up u0 into the x's.
I multiply them by the
lambdas, and then I
put them all together.
I reconstruct uk.
I hope you like that.
This formula, which it's
like common sense formula,
is exactly what that algebra
formula, matrix formula, says.
OK.
I have to do an example.
Let me finish with an example.
OK.
Here's a matrix example.
A equals-- this'll
be a special matrix.
I'm going to make the
first column add up to 1,
and I'm going to make the
second column add up to 1.
And I'm using positive numbers.
They're adding to 1.
And that's called
a Markov matrix.
So it's nice to know that
name-- Markov matrix.
One of the beauties
of linear algebra
is the variety of matrices--
orthogonal matrices,
symmetric matrices.
We'll see more and
more kinds of matrices.
And sometimes they're
named after somebody
who understood that
they were important
and found their
special properties.
So a Markov matrix is a matrix
with the columns adding up
to 1 and no negative numbers
involved, no negative numbers.
OK.
That's just by the way.
But it tells us something
about the eigenvalues here.
Well, we could find
those two eigenvalues.
We could do the determinant.
You remember how to
find eigenvalues.
The determinant of lambda I
minus A will be something.
Could easily figure it out.
There's always a lambda squared,
because it's two by two,
minus the trace.
0.8 and 0.7 is 1.5 lambda,
plus the determinant.
0.56 minus 0.06 is 0.50, 0.5.
And you set that to 0.
And you get a result that
one of the eigenvalues
is-- this factors into lambda
minus 1, lambda minus 1/2.
And the cool fact
about Markov matrices
is lambda equal 1 is
always an eigenvalue.
So lambda equal 1
is an eigenvalue.
Let's call that lambda 1.
And lambda 2 is an
eigenvalue, and that
depends on the numbers,
and it's 1/2, 0.5, 0.5.
Those are the eigenvalues.
1 plus 1/2 is 1.5.
The trace is 0.8 plus 0.7, 1.5.
Are we good for those
two eigenvalues?
Yes.
And then we find the
eigenvectors that go with them.
I think that this eigenvector
turns out to be 0.6, 0.4.
I could check.
If I multiply, I get
0.48 plus 0.12 is 0.60,
and that's the same as 0.6.
And that goes with eigenvalue 1.
And I think that this
eigenvector is 1, minus 1.
Maybe that's always for a
two-by-two Markov matrix.
Maybe that's always
the second eigenvector.
I think that's probably good.
Right.
OK.
Yeah.
All right.
What now?
What now?
I want to use the
eigenvalues and eigenvectors,
and I'm going to
write out now uk.
So if I apply A k
times to u0, I get uk.
And that's c1 1 to
the k-- this lambda 1
is 1-- times its eigenvector
0.6, 0.4 plus c2,
however much of the
second eigenvector
is in there, times
its eigenvalue,
1/2 to the k-th power
times its eigenvector,
the second eigenvector,
1, negative 1.
That is a formula.
c1 lambda 1 to the k-th
power x1 plus c2 lambda 2
to the k-th power x2.
And c1 and c2 would
be determined by u0,
which I haven't picked a u0.
I could.
But I can make the
point, because the point
I want to make is true for
every u0, every example.
And here's the point.
What happens as k gets large?
What happens if Markov
multiplies his matrix over
and over again, which is what
happens in a Markov process,
a Markov process?
This is like-- actually,
the whole Google algorithm
for page rank is based
on a Markov matrix.
So that's like a
multi-billion-dollar company
that's based on the
properties of a Markov matrix.
And you repeat it and repeat it.
That just means that Google
is looping through the web,
and if it sees a website more
often, the ranking goes up.
And if it never sees
my website, then
for that, when it was
googling some special subject,
it never came to your website
and mine, we didn't get ranked.
OK.
So this goes to 0.
1/2 to the-- it goes
fast to 0, quickly to 0.
So that goes to 0.
And of course, that stays
exactly where it is.
So there's a steady state.
What happens if page rank had
only two websites to rank,
if Google was just
ranking two websites?
Then its initial ranking,
they don't know what it is.
But by repeating the
Markov matrix and this part
going to 0, right, goes
to 0 because of 1/2
to the k-th power, there
is the ranking, 0.6, 0.4.
That's where Google-- so
this first website would
be ranked above the second one.
OK.
There's an example of a process
that's repeated and repeated,
and so a Markov matrix comes in.
This business of adding up to
1 means that nothing is lost.
Nothing is created.
You're just moving.
At every step, you
take a Markov step.
And the question is,
where do you end up?
Well, you keep moving,
but this vector
tells you how much of
the time you're spending
in the two possible locations.
And this one goes to 0.
OK.
Powers of a matrix,
powers of a Markov matrix.
Thank you.
