Question is fron subject- Linear System Theory.Need help on part (d), (e) and (f) of the problem. part (a), (b) and (c) are already solved. Problem 2 Functions of a matrix -Raxn 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)".c)Let A = QJQ-1 be any matrix decomposition. Prove f(A) = Qf(J)Q-1.eA+B = e^B¸ Hint: if two functions satisfy the same differential equation, then the uniquenessof solution of differential equations says they are equalProve det(e4) = etr(4). Note: you can use knoum facts about determinant and trace.For the time invariant linear state equation I(t) = Ar(t) show that given anyf)10 there erists a constant a(ro) such thatdet[1(t) Ar(t) ... A"'r(t)| = a(xa)e"(A)4

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Description: Question is fron subject- Linear System Theory.Need help on part (d), (e) and (f) of the problem. part (a), (b) and (c) are already solved. Problem 2 Functions of a matrix -Raxn Suppose AB = BA.Let f, g be functions over matrices and A, B Ea)Prove f(

d) To Prove eA+B=eA.eB e) To Prove deteA=etrA

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A+B - A". Hint: if two functions salisfy the same differential equation, then the uniqueness of solution of differential equations says they are equal Prove det(e^) = er(A), Note: you can use knoun facts about determinant and trace.

A+B - A". Hint: if two functions salisfy the same differential equation, then the uniqueness of solution of differential equations says they are equal Prove det(e^) = er(A), Note: you can use knoun facts about determinant and trace.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question is fron subject- Linear System Theory.

Need help on part (d), (e) and (f) of the problem. part (a), (b) and (c) are already solved.

Problem 2 Functions of a matrix -
Raxn Suppose AB = BA.
Let f, g be functions over matrices and A, B E
a)
Prove f(A)g(B) = g(B)f(A).
b)
Prove f(A") = f(A)".
c)
Let A = QJQ-1 be any matrix decomposition. Prove f(A) = Qf(J)Q-1.
eA+B = e^B¸ Hint: if two functions satisfy the same differential equation, then the uniqueness
of solution of differential equations says they are equal
Prove det(e4) = etr(4). Note: you can use knoum facts about determinant and trace.
For the time invariant linear state equation I(t) = Ar(t) show that given any
f)
10 there erists a constant a(ro) such that
det[1(t) Ar(t) ... A"'r(t)| = a(xa)e"(A)4

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