Consider the symmetric matrix ГО A = 2 L2 202 220 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=A¹¹, decompose the quadratic form Q(x) = x B x with x= [x₁x₂x₂] 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 ГО A = 2 L2 202 220 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=A¹¹, decompose the quadratic form Q(x) = x B x with x= [x₁x₂x₂] 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 !)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 7AEXP
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![Consider the symmetric matrix
ΤΟ
= 12
L2
a.
A =
2 2
02
20
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!)
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=A¹¹, decompose the quadratic form Q(x) = x B x with
x= [x₂_x₂ x₂] 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%2F0d70c926-c829-4614-add0-a4ddb63a2c02%2Ff0ak2q_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the symmetric matrix
ΤΟ
= 12
L2
a.
A =
2 2
02
20
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!)
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=A¹¹, decompose the quadratic form Q(x) = x B x with
x= [x₂_x₂ x₂] 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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