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# 1 Linear Algebra Let A be a real, symmetric, n x n matrix. Prove the following (a)-(c) using the Spectral theorem: (a) A can be expressed as ZAZ-1...

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1 Linear AlgebraLet A be a real, symmetric, n x n matrix. Prove the following (a)-(c) using the Spectral theorem:. (a) A can be expressed as ZAZ-1 with some n x n matrix Z with full rank and A being adiagonal matrix.. (b) tr (A) = E"_1 Ajj where Ajj is the (j, j) element of A.. (c) If A is positive definite, it may be written in the form A = PP', where P isn x n withrank (P) = n.Let B be a real, symmetric, idempotent, n x n matrix. Prove the following (d)-(f) using theSpectral theorem:. (d) B can be expressed as ZAZ-1 with some n x n matrix Z with full rank and A being adiagonal matrix. Furthermore, A's element is either 1 or 0.. (e) rank (B) = tr (B).. (f) In - B is also symmetric and idempotent.. (g) Let A be an x n matrix and x be an x 1 vector. Prove that:Vx (x'Ax) = x' (A + A') .If X is non-null matrix (cf. a null matrix is a matrix such that all of its elements are zero) of ordern x k, answer the following (h)-(j):. (h) Prove X'X is symmetric.. (i) Prove X'X is positive semidefinite.. (j) Provide a condition on X that makes X'X positive definite.