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.
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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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 e
a)
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 uniqueness
of 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 any
f)
xo there exists a constant a(xo) such that
det[x(t) Ax(t) ... A"-'x(t)] = a(xo)e"
Tr(A)t](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9947224-5fba-4488-8402-388e5c0c3093%2F5323ce42-7e1f-47dd-a73b-f2629e331563%2Fdm1z1or_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 2 Functions of a matrix -
R"xn. 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)".
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 uniqueness
of 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 any
f)
xo there exists a constant a(xo) such that
det[x(t) Ax(t) ... A"-'x(t)] = a(xo)e"
Tr(A)t
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