DAVID SHIROKOFF: Hi everyone.
Welcome back.
So today we're going to tackle a problem in complex matrices.
And specifically, we're going to look at diagonalizing a
complex matrix.
So given this matrix A, we're asked to find its eigenvalue
matrix lambda, and its eigenvector matrix S.
And one thing to note about this matrix A is that if we
take its conjugate transpose, it's actually equal to itself.
So in Professor Strang's book, he combines this notation to
be superscript H to mean conjugate transpose.
So if you were to take the transpose of this matrix and
then conjugate all the elements, you would find that
A equals its conjugate transpose, and we call this
property Hermitian.
So I'll let you think about this for a moment and I'll be
back in a second.
OK, welcome back.
So what's the first step in computing the eigenvectors and
eigenvalues of the matrix is to take a look at the
characteristic equation.
So specifically, we take det of A minus lambda i.
And quite possibly, the only thing new with this problem is
that the entries of the matrix A are complex.
Now, you may have already seen that lambda's being complex,
but we're going to work this out explicitly.
So if I take the determinant, we get det of 2 minus lambda,
3 minus lambda, we have 1 minus i, 1 plus i.
We want to set this to 0.
This then gives us a polynomial for lambda.
1 plus i, 1 minus i, set it equal to 0.
We can expand out this term.
You get 6 minus 5 lambda plus lambda squared.
These two terms you'll note are complex
conjugates of each other.
This tends to make things simple.
So we have minus 1 minus i squared is going to give us 2.
Because they're differences of squares, the cross terms
involving i cancel, and we get the characteristic equation.
Lambda squared minus 5 lambda plus 4 equals 0.
And specifically, we can factorize this equation.
We see that there's roots of minus 1, or factorizes into
factors of lambda -1 and lambda -4, which then give us
roots of lambda is 1 and lambda is 4.
So when one curious point to note is that the eigenvalues
are real in this case.
1 and 4 are real, whereas the matrix that we started with
was complex.
And this is a general property of Hermitian matrices.
So even though they might be complex matrices, Hermitian
matrices always have real eigenvalues.
So this is the first step when asked to diagonalize a matrix.
The second step is to find the eigenvectors.
And to do that what we have to do is we have to look at the
cases for lambda equals 1 and lambda is equal 4 separately.
So let's first look at the case of lambda is equal to 1.
And specifically, we're going to be looking for a vector
such that A minus lambda i times the vector v is 0.
And if we've done things properly, this matrix A minus
lambda i should be singular.
So if we take A minus lambda i, we're going to get 1 1
minus i, 1 plus i 3 minus 1 is 2.
And I'll write out components of v, which are v1 and v2.
And we want this to be 0.
And you'll note that it's almost always the case that
when we work out A minus lambda i, the second row is
going to be a constant multiple of the first row.
And this must be the case because these two rows must be
linearly dependent on each other for the matrix A minus
lambda i to be singular.
So if you look at this you might think that these two
rows aren't necessarily linearly independent.
But the point is that there's complex numbers involved.
And indeed, actually if we were to multiply this first
row by 1 plus lambda, we would get 1 plus lambda and 2.
And you note that that's exactly the second row.
So this second row is actually 1 plus lambda
times the first row.
So these rows are actually linearly
dependent on each other.
So what values of v1 and v2 can we take?
Well we just need to make this top row multiplied by v1 and
v2 equal to 0.
And then because the second row is a constant multiple of
the first row, we're automatically guaranteed that
the second equation holds.
So just by looking at it, I'm going to take v1 is equal to 1
minus i, and v2 is negative 1.
So we see that 1 times 1 minus i minus 1 times 1 minus i is
going to give us 0.
So this is one solution.
And of course, we can take any constant multiple times this
eigenvector, and that's also going to be an eigenvector.
So I'll just write this out.
1 minus i, minus 1 is the eigenvector for lambda is
equal to 1.
For lambda is equal to 4, again, A minus lambda i, is
going to give us negative 2, 1 minus i, 1 plus i, 3 minus
lambda's going to be minus 1.
And I'll call this vector u1 and u2.
And again, we want u1 and u2 equal to 0--
or sorry, the matrix multiplied by u1,
u2 is equal to 0.
And just by looking at this again, we see that the second
row is actually a constant multiple of the first row.
For example, if we were to multiply this row by negative
2, and this row by 1 plus i, we would see that they're
constant multiples of each other.
So I can take u to be, for example, 1, and 1 plus i.
How did I get this?
Well I just looked at the second equation because it's a
little simpler, and I said well if I have 1 plus I here,
I can just say multiply it by 1.
And then minus 1 times 1 plus I when I add them
up is going to vanish.
So this is how I get the second one.
Now there's something curious going on, and this is going to
be another property of Hermitian matrices.
But if you actually take a look at this eigenvector, it
will be orthogonal to this eigenvector when we conjugate
the elements and dot the two vectors together.
So this is another very special property of complex
Hermitian matrices.
OK, so the last step now is to construct these matrices
lambda and S. Now we already know what lambda is because
it's the diagonal matrix with the eigenvalues 1 and 4.
So we have 1 0, 0 and 4.
Now I'm going to do something special for S. I've noted that
these two vectors u and v are orthogonal to each other.
So what do I mean by orthogonal?
Specifically, if I were to take v conjugate transpose and
multiply it by u, we would end up getting 1 plus i minus 1.
This would be v conjugate transpose.
1, 1 plus i, and we see that when we multiply
these out we get 0.
So when we have orthogonal eigenvectors, there's a trick
that we can do to build up this matrix S and S inverse.
What we can do is we can normalize u and v. So
specifically, we can take any constant multiple of u and v,
and it's still going to be an eigenvector.
So what I'm going to do is I'm going to take u and v and
multiply them by their length.
So for example u, the amplitude of its top
component is 1.
The amplitude of its bottom component is 2.
So notice that the modulus of the complex
number 1 plus I is 2.
So we have--
sorry, it's root 2, the complex modulus is root 2.
So the amplitude of the entire vector is root 3.
It's 1 plus 2 squared quantity squared rooted,
so it's root 3.
So what we can do is we can build up this matrix S using a
normalization factor of 1 over root 3.
And I'm going to take the first column is the first
eigenvector that corresponds to eigenvalue 1.
And then the second column is the second eigenvector which
corresponds value 4.
And the reason I put in this root 3 here is to make this
column unit length 1, and this column unit length 1.
And the reason I do this is because now this matrix S,
it's possible to check that this matrix S is actually
unitary, which means that its inverse is actually just equal
to it's conjugate transpose.
So this is a very special property of the eigenvectors
of a Hermitian matrix.
And then lastly, I'm just going to write down the
diagonalization of A. So if I have A, because I have its
eigenvector matrix S, and its eigenvalue matrix lambda, it's
possible to decompose A into a product
of S lambda, S inverse.
And because S is unitary, its inverse is actually its
conjugate transpose.
So just to put the pieces together, we have A is equal
to S, which is 1 over root 3.
1 minus i, minus 1, 1, 1 plus i times the diagonal matrix 1
0, 0 4 times S inverse, which is going to be
its conjugate transpose.
So what I do is I conjugate each element, so 1 minus i
becomes 1 plus i and vice versa.
And then I take the transpose.
So I get 1 plus i.
Transposing swaps the minus 1 and 1.
And at the end of the day, I get S inverse is just this
matrix here.
And if you were to multiply these matrices out, you would
see it you actually do recover A.
So just to summarize quickly, even though we were given a
complex matrix A, the process to diagonalize A is the same
as what we've seen before.
The first step is to find the characteristic equation and
the eigenvalues.
And then the second step is to find the eigenvectors, and you
do this in the same procedure.
But in general, the eigenvectors can be complex.
And for this very special case, when A is Hermitian, the
eigenvalues are real, and the eigenvectors are orthogonal to
each other.
So I think I'll conclude here, and I'll see you next time.