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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.

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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.1: Introduction To Eigenvalues And Eigenvectors
Problem 35EQ
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Consider the matrix A = 2 | 0 a. 1 2 -1 2 4 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) 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 B = (A+A¹)´¹ is positive definite, negative definite or indefinite, without computing its eigenvalue decomposition. (Hint: use the elimination method and Hermite's theorem).
Transcribed Image Text:Consider the matrix A = 2 | 0 a. 1 2 -1 2 4 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) 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 B = (A+A¹)´¹ is positive definite, negative definite or indefinite, without computing its eigenvalue decomposition. (Hint: use the elimination method and Hermite's theorem).
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