Hermitian matrix

1. Oct 13, 2007

sunnyo7

Its quantum computing but related to math:
1. The problem statement, all variables and given/known data

show every hermitian matrix can be diagonalized by unitary matrix. Prove this using. N x N matrix.

2. Relevant equations
H= hermitian matrix. U = unitary matrix
show U-1(inverse)HU = D (diagonal) using N x N matrix.

3. The attempt at a solution
I don't know how to start.

2. Oct 13, 2007

ZioX

This is fairly nontrivial, but easy if you know what to do. It is a special case of the finite dimensional spectral theorem, as hermitian matrices are normal.

Sketch: Suppose A is normal. Then by Schur's Lemma we know that A is unitarily similar to an upper triangular matrix. That is, $$U^HAU=T$$.

Now, since T is both triangular and normal we see that T is diagonal. Hence, $$A=UDU^H$$ upon block multiplication.

I will leave it to you to show that T is normal and if T triangular and normal then T is diagonal.

Last edited: Oct 13, 2007
3. Oct 13, 2007

sunnyo7

No, still not clear. I'm good with kind of proving. Can you be more specific on how to answer the question. I'm really really thankfu.
Question:
It is possible to show using the Spectral Decomposition theorem that every hermitian matrix cab be diagonalzed by a unitary matrix. Let H be a an hermitian matrix and set S be unitary where S(inverse)HS= D where D is diagonal.
Show that S(inverse) H S and hence D is hermitian