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Consider the matrix A = 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 ex- planation!) 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 = 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 ex- planation!) 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 0 2 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 ex- planation!) 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 + AT) is positive definite, negative definite, or indefinite, with- out computing its eigenvalue decomposition. (Hint: decompose the quadratic form Q(x) = x² (A+A¹)x as the sum of squares of independent linear forms using the elimination method and use Hermite's theorem.)
Transcribed Image Text:Consider the matrix A = 1 0 2 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 ex- planation!) 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 + AT) is positive definite, negative definite, or indefinite, with- out computing its eigenvalue decomposition. (Hint: decompose the quadratic form Q(x) = x² (A+A¹)x as the sum of squares of independent linear forms using the elimination method and use Hermite's theorem.)
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