# Hermitian matrix

sunnyo7
Its quantum computing but related to math:

## Homework Statement

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

## Homework Equations

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

## The Attempt at a Solution

I don't know how to start.

Any help would be helpful.

## Answers and Replies

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:
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