# Unitary Matrices Proof

Problem:

Let $$A, B \in \mathbb{C}^{n\times n}$$ be selfadjoint ,such that $$[A,B] := AB − BA = 0$$ Show that there is a unitary matrix $$U \in \mathbb{C}^{n\times n}$$ such that $$U^*$$A$$U$$ and $$U^*BU$$ are both diagonal.

I am very confused about this problem. Should I first prove that $$U$$ is unitary and then that $$U^*$$A$$U$$ and $$U^*$$B$$U$$ are both diagonal? Or I should assume that $$U^*$$A$$U$$ and $$U^*$$B$$U$$ are diagonal and therefoe I should prove that $$U$$ is unitary??

Hint: Since $$A:\mathbb{C}^n\to\mathbb{C}^n$$ is self-adjoint, we can find an orthonormal basis composed of eigenvectors of $$A$$ by spectral theorem. Thus we can decompose $$\mathbb{C}^n = \bigoplus_{i=1}^k \ker(A-\lambda_iI).$$ Let $$N_i = \ker (A-\lambda_i)$$ and show that $$B_i =B|_{N_i}:N_i\to N_i$$ is a well-defined self-adjoint operator. Find an orthonormal basis $$\mathcal{B}_i$$ of $$N_i$$ consisting of eigenvectors of $$B_i$$ for each $$i$$. Finally prove that $$\mathcal{B}=\bigcup_{i=1}^k \mathcal{B}_i$$ forms an orthonormal basis of $$\mathbb{C}^n$$ consisting of eigenvectors of both $$A$$ and $$B$$. And form a desired unitary matrix $$U$$ using the orthonormal basis $$\mathcal{B}$$.