# Linear Algebra

1. Sep 27, 2006

### pivoxa15

Find the diagonal form of the Hermitian matrix

$$A=\left( \begin{array}{cc} 2 & 3i\\ -3i & 2 \end{array} \right)$$

The spectral theorem could be used with PAP*=D where D is diagonal matrix and P is a unitary matrix.

I put the columns of P as the eigenvectors (with unit length) of A,

$$P=\frac{1}{\sqrt{2}}\left( \begin{array}{cc} i & -i\\ 1 & 1 \end{array} \right)$$

I have checked that P is unitary with $$P^{-1}=P^{*}$$ and the diagonal entries of D should be 5 and -1. But I got

$$D=\left( \begin{array}{cc} 2 & -3\\ -3 & 2 \end{array} \right)$$

which clearly isn't correct.

Last edited: Sep 27, 2006
2. Sep 28, 2006

### matt grime

Yes, you're right it isn't correct. I'm, nots sure what you want anyone here to do. You have the method correct, so just make sure you'renot making any dumb mistakes in multiplying out matrices.

3. Sep 28, 2006

### greisen

hey
rigth method but wrong eigenvalues

P = 1/sqrt(2) [i -1;i 1]

this will help

4. Sep 28, 2006

### Hurkyl

Staff Emeritus
I can never remember for sure, but isn't it supposed to be A = PDP*? (and thus P*AP = D?)

5. Sep 28, 2006

### pivoxa15

That could be my mistake.