Consider the matrix 1 A = 1 - 1 LO 0 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+AT) is positive definite, negative definite or indefinite, without computing its eigenvalue decomposition. T 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.
Consider the matrix 1 A = 1 - 1 LO 0 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+AT) is positive definite, negative definite or indefinite, without computing its eigenvalue decomposition. T 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.
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
1
A = 1
-
1
LO
0
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+AT) is positive definite, negative definite or indefinite, without
computing its eigenvalue decomposition.
T
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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