00:01
Okay, so let's take some hermission matrix a, so this just means that a is equal to its own adjoint, and we want to show that the eigenvalues must be real.
00:11
So let's let lambda be an eigenvalue, and let's take the vector v to be the corresponding eigenvector.
00:22
We're going to look at the inner product of v with av.
00:27
So the thing to notice here is that v is not zero also.
00:31
It can't be the zero vector.
00:33
Will be important later on.
00:35
So if we look at the inner product of of v with a v, well, on the one hand, this inner product is equal to v with lambda v because by definition, av is equal to lambda v, since v is an eigenvector with eigenvalue lambda.
00:56
Now, if we take out this lambda, we get lambda conjugate inner product of v with v.
01:05
So this is one result...