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MG1. In this problem, we will prove some facts about a special class of complex matrices. Let F = C, the complex field. - A Hermitian matrix A is a square complex matrix with the property A = A (A is the conjugate transpose of A), - a unitary matrix U is a square complex matrix with the property U . Ut = Ut . U = In where In is the n x n identity matrix, - an upper triangular matrix B = (bij) is a square complex matrix such that bij = 0 whenever i > j, - a diagonal matrix D = (dij) is a square complex matrix such that dij = 0 whenever if j. The questions are: (a) Prove that the multiplication of two Hermitian matrices is a Hermitian matrix. (b) We have Schur's Lemma (you don't have to prove it): Let A be any n x n complex matrix. Then there exist an n x n upper triangular matrix B and an n x n unitary matrix U such that A = U . B . U. Use Schur's Lemma and (a) to prove the following statement: Let A be an n x n Hermitian matrix. There exist an n x n diagonal matrix D and an n x n unitary matrix U such that A = U . D . U. (c) Let V be a finite-dimensional complex inner product space and T be a linear operator on V. Show that T is self-adjoint if and only if the represent matrix of T is a Hermitian matrix with respect to any orthonormal basis of V