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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