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(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P such that PT AP is a diagonal matrix Hint : UseLO and Problem EK〗 (g) Let A be a square matrix and Rn × Rn → Rn is defined by: UCTION E AND MES FOR THE la(x, y) = хтАУ (i) Show that I is symmetric, ie, 14(x,y) = 1a(y, x), if a d Only if. A is symmetric (ii) Show that for a symmetric matrix A, the eigenvalues of A are all positive, if and only if, ER(S) AND T DETAILS CES T SUPPORT ES LAN AL WORK AND NTEGRATED la(x, x) > 0, with equality precisely for x = 0 INNER PRODUCTS AND MATRICES A bilinear function V × V → R is said to be positive definite if for each x e V, x 0, MENT NTLY ASKED ONS ING 34 DUM A: MENTS FIRST ER DUM B: MENTS SECOND ER DUM C: USEFUL TER SOFTWARE DUM D: TARY LINEAR A USING MAXIMA MAT2611/101/3/2019 It follows from(c& gthat real symmetric matrices with positive eigenvalues precisely correspond to inner products, which are symmetric positive definite bilinear functions Mark Distribution for Problem 3 5 +55+5(3 (2 +3+2))5(22)(3+3+3+2)) 50 marks)
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