I thought, S is self-adjoint because S^∗ = (A ^∗A) = A ^∗ (A^∗ ) ^∗ = A^∗A = S. Therefore, spectral theorem ⇒ there is an orthonormal basis {e1, ..., en} of eigenvectors of S. But I guess I'm far from the solution. I would be very happy if you have a solution suggestion or solution.
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Question;
Over the complex numbers we consider the matrix
Find a unitary matrix S ∈ C ^ 4×4 such that S ∗AS is diagonal. A matrix S ∈ C^ n×n is said to be unitary if S ∗ = S¯^T = S ^−1.