Question

a 2×2 matrix is both symmetric and orthogonal but not symmetric .find eigen values of the matrix

Answer 1

In general, if AA is symmetric, it is orthogonally diagonalisable and all its eigenvalues are real. If it is also orthogonal, its eigenvalues must be 1 or -1. It follows that every symmetric orthogonal matrix is of the form QDQ⊤QDQ⊤, where QQ is a real orthogonal matrix and DD is a diagonal matrix whose diagonal entries are 1 or -1.