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Consider the symmetric matrix 0 −1 1 69 0 -1 1 -1 0 A = a) Diagonalize the matrix A in the form A = SAST, with S an orthogonal matrix containing the (normalized) eigenvectors and A a diagonal matrix containing the eigenvalues. b) Using the eigenvalue decomposition computed in a), determine (including a short explanation!) a. the rank of the matrix A.

Consider the symmetric matrix 0 −1 1 69 0 -1 1 -1 0 A = a) Diagonalize the matrix A in the form A = SAST, with S an orthogonal matrix containing the (normalized) eigenvectors and A a diagonal matrix containing the eigenvalues. b) Using the eigenvalue decomposition computed in a), determine (including a short explanation!) a. the rank of the matrix A.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.5: Applications
Problem 74EQ
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Consider the symmetric matrix A = C. 0 -1 1 -1 1 0 -1 0 -1 a) Diagonalize the matrix A in the form A = SA ST, with S an orthogonal matrix containing the (normalized) eigenvectors and A a diagonal matrix containing the eigenvalues. b) 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. the null space of the matrix A. c) Decompose the quadratic form Q(x) = x¹ B x with B = A² and x=[x₁x₂x₂] as the sum of r = rank(B) squares of independent linear forms. (Note: different solutions exist, one is sufficient! Either use the elimination method or the eigenvalue decomposition computed in a) where you don't need to explicitly compute B).
Transcribed Image Text:Consider the symmetric matrix A = C. 0 -1 1 -1 1 0 -1 0 -1 a) Diagonalize the matrix A in the form A = SA ST, with S an orthogonal matrix containing the (normalized) eigenvectors and A a diagonal matrix containing the eigenvalues. b) 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. the null space of the matrix A. c) Decompose the quadratic form Q(x) = x¹ B x with B = A² and x=[x₁x₂x₂] as the sum of r = rank(B) squares of independent linear forms. (Note: different solutions exist, one is sufficient! Either use the elimination method or the eigenvalue decomposition computed in a) where you don't need to explicitly compute B).
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