Consider the symmetric matrix ΓΟ 2 2 022 A = 2 L2 220 202 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. b. the determinant of the matrix A. c. the null space of the matrix A. c) For the symmetric matrix B=A 1, decompose the quadratic form Q(x) = x B x with x= [× × ×3] as the sum of r = rank(B) squares of independent linear forms. (Note: different solutions exist, one is sufficient. Using the eigenvalue decomposition computed in a), this question can even be answered without computing the inverse matrix !)
Consider the symmetric matrix ΓΟ 2 2 022 A = 2 L2 220 202 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. b. the determinant of the matrix A. c. the null space of the matrix A. c) For the symmetric matrix B=A 1, decompose the quadratic form Q(x) = x B x with x= [× × ×3] as the sum of r = rank(B) squares of independent linear forms. (Note: different solutions exist, one is sufficient. Using the eigenvalue decomposition computed in a), this question can even be answered without computing the inverse matrix !)
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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![Consider the symmetric matrix
ΓΟ 2 2
022
A = 2
L2
220
202
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.
b. the determinant of the matrix A.
c. the null space of the matrix A.
c) For the symmetric matrix B=A 1, decompose the quadratic form Q(x) = x B x with
x= [× × ×3] as the sum of r = rank(B) squares of independent linear forms.
(Note: different solutions exist, one is sufficient. Using the eigenvalue decomposition computed
in a), this question can even be answered without computing the inverse matrix !)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6070c27-0824-4884-a918-195c8f609349%2F4d30344f-dca8-4bce-afbc-9f16580b495c%2Fy30b2fo_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the symmetric matrix
ΓΟ 2 2
022
A = 2
L2
220
202
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.
b. the determinant of the matrix A.
c. the null space of the matrix A.
c) For the symmetric matrix B=A 1, decompose the quadratic form Q(x) = x B x with
x= [× × ×3] as the sum of r = rank(B) squares of independent linear forms.
(Note: different solutions exist, one is sufficient. Using the eigenvalue decomposition computed
in a), this question can even be answered without computing the inverse matrix !)
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