Consider the symmetric matrix 0 -1 1 A = 0 −1 1 -1 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 rank of the matrix A. b. the determinant of the matrix A. C. the null space of the matrix A. c) Decompose the quadratic form Q(x) = x B x with B = A² and x = r = rank(B) squares of independent linear forms. = [x₁ X2 x3] as the sum of (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).
Consider the symmetric matrix 0 -1 1 A = 0 −1 1 -1 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 rank of the matrix A. b. the determinant of the matrix A. C. the null space of the matrix A. c) Decompose the quadratic form Q(x) = x B x with B = A² and x = r = rank(B) squares of independent linear forms. = [x₁ X2 x3] as the sum of (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).
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
0
-1
1
A =
0 −1
1
-1 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 rank of the matrix A.
b. the determinant of the matrix A.
C. the null space of the matrix A.
c) Decompose the quadratic form Q(x) = x B x with B = A² and x =
r = rank(B) squares of independent linear forms.
= [x₁
X2
x3] as the sum of
(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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6070c27-0824-4884-a918-195c8f609349%2F5d58fbcb-c5cd-41c5-9acd-4f57256af644%2F613m2bvb_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the symmetric matrix
0
-1
1
A =
0 −1
1
-1 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 rank of the matrix A.
b. the determinant of the matrix A.
C. the null space of the matrix A.
c) Decompose the quadratic form Q(x) = x B x with B = A² and x =
r = rank(B) squares of independent linear forms.
= [x₁
X2
x3] as the sum of
(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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