Diagonal matrix is normal

Question

Suppose $A$ is a diagonal matrix. I want to show that $A$ is normal. That is: $A^* A = A A^* $. Since $A$ is diagonal, then $A = A^*$ and so $A$ is trivially normal. IS this correct? or I am missing something?

Answer 1

If $A$ is real, then you're right.

If $A$ is complex, then $A$ is not the same as $A^*$.

In any case, $A^* A = A A^*$ because $A^* A$ and $A A^*$ are diagonal matrices with entries $\overline a_{ii} a_{ii}=a_{ii} \overline a_{ii}$.

Answer 2

It is not true that $A=A^{\ast}$ is general; this is true if and only if $A$ is real.

In multiplying diagonal matrices all that happens is multiplying the diagonal elements pairwise. This is obviously commutative, so multiplication of diagonal matrices is commutative.