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Consider the matrix A = 5 1 0 5 1 0 l-1 1 6 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) Is the matrix S an orthogonal matrix ? Why / why not? c) 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.

Consider the matrix A = 5 1 0 5 1 0 l-1 1 6 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) Is the matrix S an orthogonal matrix ? Why / why not? c) 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.

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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Consider the matrix 1 1 −1 1 6 5 A = 5 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. a. b) Is the matrix S an orthogonal matrix? Why / why not? c) Using the eigenvalue decomposition computed in a), determine (including a short explanation!) the rank of the matrix A. b. the determinant of the matrix A. C. the null space of the matrix A. d) Determine if the matrix (A+A) is positive definite, negative definite or indefinite, Hint: this can be determined without computing the eigenvalue decomposition.
Transcribed Image Text:Consider the matrix 1 1 −1 1 6 5 A = 5 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. a. b) Is the matrix S an orthogonal matrix? Why / why not? c) Using the eigenvalue decomposition computed in a), determine (including a short explanation!) the rank of the matrix A. b. the determinant of the matrix A. C. the null space of the matrix A. d) Determine if the matrix (A+A) is positive definite, negative definite or indefinite, Hint: this can be determined without computing the eigenvalue decomposition.
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