# Determinant of a Hermitiain matrix

1. Nov 20, 2007

### kingwinner

Q: Suppose A is a Hermitiain matrix, prove that det A is real.

Note: I know nothing about inner products yet.

Some thoughts:
Perhaps proving that det A = conjugate of (det A) ? But how?

2. Nov 20, 2007

### morphism

Try proving that det(conjugate of A) = conjugate of det(A) (use the formula for det(.)).

3. Nov 20, 2007

### kingwinner

How can I do that? Can you please tell me more about it?

4. Nov 21, 2007

### Kreizhn

I'm not terribly sure as to what formula morphism is referring to. There is a process/algorithm for finding the determinant of a general matrix but no closed formula.

You know that by the definition of a Hermitian Matrix, it's equal to it's transpose conjugate. Furthermore, we know that the the determinant of the transpose is equal to the determinant of the original matrix. Thus using this bit of information, this is a one line proof.

5. Nov 21, 2007

### kingwinner

But this just gives det A = det (conjugate of A)

6. Nov 21, 2007

### morphism

7. Nov 21, 2007

### Kreizhn

It may be considered a formula semantically in that following the outlined process will give you the determinant, but I don't believe that we can consider it a formula in the classical sense. Rather it's a procedure, which logically is a very different thing.

Furthermore, the determinant of the conjugate is the conjugate of the determinant.

8. Nov 21, 2007

### morphism

Yes, and using that formula (or procedure - call it whatever you want) gives a very easy proof of that, like I stated in post #2.