I am given a matrix $A$. I find out it is normal. And I compute $\det(A-\lambda)=0$ and find that not all $\lambda_i$ are different, i.e., the eigenvalues are not distinct. Thus, I am not sure if the eigenvectors are L.I.
Now, I want find a decomposition $$A=UDU^H$$ but how do I do it? And under what conditions can I do that?
- $D$ is a diagonal matrix
- $^H$ is the hermitian conjucate
- $U$ is unitary matrix.