Consider the matrix 202 -1 21 T41 202 A = 0 a) Diagonalize the matrix in the form A = SAS-1, with S a matrix containing the (normalized) eigenvectors and A a diagonal matrix containing the eigenvalues. b) Using the eigenvalue decomposition computed in a), determine (including a short explanation!) a. the rank of the matrix A. b. the determinant of the matrix A. c. the null space of the matrix A. c) Determine if the matrix (A+AT) is positive definite, negative definite or indefinite. Hint: this can be determined without computing the eigenvalue decomposition (e.g. using properties of the eigenvalues and/or using Hermite's theorem).
Consider the matrix 202 -1 21 T41 202 A = 0 a) Diagonalize the matrix in the form A = SAS-1, with S a matrix containing the (normalized) eigenvectors and A a diagonal matrix containing the eigenvalues. b) Using the eigenvalue decomposition computed in a), determine (including a short explanation!) a. the rank of the matrix A. b. the determinant of the matrix A. c. the null space of the matrix A. c) Determine if the matrix (A+AT) is positive definite, negative definite or indefinite. Hint: this can be determined without computing the eigenvalue decomposition (e.g. using properties of the eigenvalues and/or using Hermite's theorem).
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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Transcribed Image Text:Consider the matrix
202
-1 21
T41
202
A = 0
a) Diagonalize the matrix in the form A = SAS-1, with S a matrix containing the (normalized)
eigenvectors and A a diagonal matrix containing the eigenvalues.
b) Using the eigenvalue decomposition computed in a), determine (including a short explanation!)
a. the rank of the matrix A.
b. the determinant of the matrix A.
c. the null space of the matrix A.
c) Determine if the matrix (A+AT) is positive definite, negative definite or indefinite.
Hint: this can be determined without computing the eigenvalue decomposition (e.g. using
properties of the eigenvalues and/or using Hermite's theorem).
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