Consider the symmetric matrix 1 1 A = 0 1 1 0. 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. 2/2 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) Compute the symmetric matrix B=A² and 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! Either use the elimination method or the eigenvalue decomposition computed in a)
Consider the symmetric matrix 1 1 A = 0 1 1 0. 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. 2/2 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) Compute the symmetric matrix B=A² and 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! 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 32E
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![Consider the symmetric matrix
1
1
A =
0
1
1
0.
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.
2/2
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) Compute the symmetric matrix B=A² and 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! Either use the elimination method or the eigenvalue
decomposition computed in a)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6070c27-0824-4884-a918-195c8f609349%2F445217df-4b61-482b-8942-2f022a59272b%2F4pkn8d9_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the symmetric matrix
1
1
A =
0
1
1
0.
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.
2/2
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) Compute the symmetric matrix B=A² and 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! Either use the elimination method or the eigenvalue
decomposition computed in a)
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