Question is from subject Linear System Theory Problem 2 Functions of a matrix -R"xn. Suppose AB = BA.Let f, g be functions over matrices and A, B ea)Prove f(A)g(B) = g(B)f(A).b)Prove f(A") = f(A)".с)Let A = QJQ-1 be any matrix decomposition. Prove f(A) = Qf(J)Q-1.d)eA+B = e^eB. Hint: if two functions satisfy the same differential equation, then the uniquenessof solution of differential equations says they are equal.e)Prove det(e4) = etr(A). Note: you can use known facts about determinant and trace.For the time invariant linear state equation x(t): Аг(t) show that given anyf)xo there exists a constant a(xo) such thatdet[x(t) Ax(t) ... A"-'x(t)] = a(xo)e"Tr(A)t

Name: Question is from subject Linear System Theory Problem 2 Functions of a
Uploaded: 2024-03-20T07:07:26.000Z
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Description: Question is from subject Linear System Theory Problem 2 Functions of a matrix -R"xn. Suppose AB = BA.Let f, g be functions over matrices and A, B ea)Prove f(A)g(B) = g(B)f(A).b)Prove f(A") = f(A)".с)Let A = QJQ-1 be any matrix decomposition. Prove f(

Given that f, g be the functions over matrix A, B∈ℝn×n and AB=BA let A and B be two 2×2 matrices…

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Problem 2 Fumctions of a matrix - Let f, g be functions over matrices and A, B e R"xn. Suppose AB = BA. a) Prove f(A)g(B) = g(B)f(A). b) Prove f(A") = f(A)". %3D c) Let A = QJQ¬1 be any matrix decomposition. Prove f(A) = Qf(J)Q-1.