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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
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.
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.
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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