Proof of a property of a matrix

Question

Is this statement true: $$HH^\dagger = H^\dagger H \implies H = H^\dagger$$

NOTE: $H^\dagger$ is the conjugate transpose of $H$

If so, can somebody provide a proof?

Answer 1

This is not true: $(-i)\cdot i = i\cdot (-i)$ but $i\ne -i$.

Answer 2

If $H$ is a diagonal matrix, then $H.H^\dagger=H^\dagger.H$, but you will only have $H=H^\dagger$ if all entries are real.

Answer 3

No this is not the case.

A matrix with this property (that it commutes with its conjugate transpose) is said to be Normal.

As an example

$$H = \left( \begin{array}{cc} 0 & -1\\ 1 & 0 \end{array}\right)$$

Then clearly $H \neq H^\dagger$, but you can show $ H H^\dagger = H^\dagger H = I$, the identity matrix.