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).
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
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).
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