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Consider the matrix 1 1 [1 A = 1 -1 1 Lo 0 1] a) Compute the eigenvalue decomposition (eigenvalues A and eigenvectors S) of the matrix A. b) Are the eigenvectors orthogonal ? 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 1 1 [1 A = 1 -1 1 Lo 0 1] a) Compute the eigenvalue decomposition (eigenvalues A and eigenvectors S) of the matrix A. b) Are the eigenvectors orthogonal ? 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 A = 1 Lo -1 0 1. a) Compute the eigenvalue decomposition (eigenvalues A and eigenvectors S) of the matrix A. b) Are the eigenvectors orthogonal? 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. d) Determine if the matrix (A+A¹) is positive definite, negative definite or indefinite, without computing its eigenvalue decomposition. Hint: decompose the quadratic form Q(x) = x² (A + A²)x as a sum of squares of independent linear forms using the elimination method and use Hermite's theorem.
Transcribed Image Text:Consider the matrix A = 1 Lo -1 0 1. a) Compute the eigenvalue decomposition (eigenvalues A and eigenvectors S) of the matrix A. b) Are the eigenvectors orthogonal? 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. d) Determine if the matrix (A+A¹) is positive definite, negative definite or indefinite, without computing its eigenvalue decomposition. Hint: decompose the quadratic form Q(x) = x² (A + A²)x as a sum of squares of independent linear forms using the elimination method and use Hermite's theorem.
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