Let $A \in M_2(\mathbb{C})$ be a hermitian matrix i.e. $A = A^*$.
Suppose that $\lambda, \mu$ are eigenvalues correspoing to normed eigenvectors $\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}, \begin{pmatrix} w_1 \\ w_2 \end{pmatrix}$.
I would like to show that $U = \begin{pmatrix} v_1 & w_1 \\ v_2 & w_2 \end{pmatrix}$ is indeed unitary.
I know that $U^* = \begin{pmatrix} \bar{v_1} & \bar{v_2} \\ \bar{w_1} & \bar{w_2} \end{pmatrix}$
Showing that $U^*U = 1_2$ is not a problem since those eigenvectors are orthonormal.
But I don't how to show that converse i.e. $UU^*$ is also an unitary matrix.
It leads me to entries of type $v_1\bar{v_1} + w_1\bar{w_1}$ and I don't know how to conclude that it is also unitary.
Best Answer
For square matrices, if $AB=I$, then $BA=I$. So you already have that $U$ is a unitary.