a) Determine the eigenvalues A1, A2 and eigenvectors V₁, V₂ of ^-( :). A = 1 b) Find transformation matrices T and T-1 and a diagonal matrix D such that T-¹AT DA TDT-¹. (Hint: T is the column matrix for a choice of eigenvectors determined in a).) Optional: Since A = A* is real and symmetric all of the eigenvectors will be orthogonal and if we normalize each column vector in the column matrix to have length/magnitude 1, then T-¹ = TT. In this context, one says that the matrix T is orthogonal. c) Compute the flow matrix/exponential matrix using exp(tA) = Texp(tD)T-¹ = T exit 0 0 e^2t T-¹.
a) Determine the eigenvalues A1, A2 and eigenvectors V₁, V₂ of ^-( :). A = 1 b) Find transformation matrices T and T-1 and a diagonal matrix D such that T-¹AT DA TDT-¹. (Hint: T is the column matrix for a choice of eigenvectors determined in a).) Optional: Since A = A* is real and symmetric all of the eigenvectors will be orthogonal and if we normalize each column vector in the column matrix to have length/magnitude 1, then T-¹ = TT. In this context, one says that the matrix T is orthogonal. c) Compute the flow matrix/exponential matrix using exp(tA) = Texp(tD)T-¹ = T exit 0 0 e^2t T-¹.
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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If someone could help me with this practice question I would be extremely appreciative. While I do know how to get the eigenvalues and eigenvectors, I'm struggling with the rest of the question.
Transcribed Image Text:Q.1. Consider the differential equation
- ( ) x + ( )
(1)
a) Determine the eigenvalues A₁, A2 and eigenvectors V₁, V₂ of
1 1
- (+)
=
1 1
b) Find transformation matrices T and T-1 and a diagonal matrix D such that
T¹AT DA
TDT-¹.
x' (t):
A
(Hint: T is the column matrix for a choice of eigenvectors determined in a).)
Optional: Since A = A* is real and symmetric all of the eigenvectors will be orthogonal and if we
normalize each column vector in the column matrix to have length/magnitude 1, then T-¹ = TT.
In this context, one says that the matrix T is orthogonal.
c) Compute the flow matrix/exponential matrix using
1
exp(tA) = Texp(tD) T-¹
x' (t)
=
=
=
T
exit
0
d) Find a fundamental set of solutions x(¹) and x(2) of the corresponding homogeneuous differential
equation
(11)
0
ed2t
X.
T-¹.
(Hint: The column of the exponential matrix determined in c) can be chosen for x(¹), x(2).)
e) Find the general solution of (1) by using the variation of parameter formula
x(t) = cx(¹)(t) + c₂x(²)(t) + exp(†A) [. exp((-t) A)g(t) dt,
where g(t) = (0, e¹)T is the nonhomogenous term in the differential equation.
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