Problem 7 (8 points). Frobenius norm and its monotonicity. Given A Rmxn, let m n...

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Problem 7 (8 points). Frobenius norm and its monotonicity. Given A € Rmxn, let m n ||A||F=Σa, i=1 j=1 known as the Frobenius norm. Further, let (A,B) F = trace (ATB), with A, B € Rmxn known as the Frobenius inner product. (a) (2 points). Prove that ||||F is a norm in Rmxn. Hint: You may use the fact that the usual Euclidean norm (1,..., #₂)|| 2₂ = √√²+...+ is indeed a norm. (b) (2 points). Prove that (,)F is an inner product in Rmxn (c) (1 points). Prove that ||A+ B||-|| A||2+2(A, B) F+ || B|| = (d) (3 points). Suppose that m = n, A, B 0, and A B (i.e., B-A0). Prove that ||A|F |BF (this is known as monotonicity of ||||F). Hint: If X0, then there exists another matrix Y Z 0 such that Y² = X. Such Y is not unique in general, but we may denote it as X¹/2 Note: Please solve (a), (b), (c), (d) in the order that they appear. Problem 7 (8 points). Frobenius norm and its monotonicity. Given A € Rmxn, let m n ||A||F=Σa, i=1 j=1 known as the Frobenius norm. Further, let (A,B) F = trace (ATB), with A, B € Rmxn known as the Frobenius inner product. (a) (2 points). Prove that ||||F is a norm in Rmxn. Hint: You may use the fact that the usual Euclidean norm (1,..., #₂)|| 2₂ = √√²+...+ is indeed a norm. (b) (2 points). Prove that (,)F is an inner product in Rmxn (c) (1 points). Prove that ||A+ B||-|| A||2+2(A, B) F+ || B|| = (d) (3 points). Suppose that m = n, A, B 0, and A B (i.e., B-A0). Prove that ||A|F |BF (this is known as monotonicity of ||||F). Hint: If X0, then there exists another matrix Y Z 0 such that Y² = X. Such Y is not unique in general, but we may denote it as X¹/2 Note: Please solve (a), (b), (c), (d) in the order that they appear.