Consider the matrix A = 1 0 0 0 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 null space of the matrix A. b. the determinant of the matrix A. c. if the matrix A is positive definite, negative definite or indefinite. c) Compute the matrix B=A¹¹ d) Decompose the quadratic form Q(x) = x B x with x = [x₁x₂x3 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).

Consider the matrix A = 1 0 0 0 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 null space of the matrix A. b. the determinant of the matrix A. c. if the matrix A is positive definite, negative definite or indefinite. c) Compute the matrix B=A¹¹ d) Decompose the quadratic form Q(x) = x B x with x = [x₁x₂x3 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).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 30E

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