Consider the quadratic form Q(x1, x2, x3) = 2x² + 2x² + 2x² + 2x1x2 + 2x1x3 + 2x2x3 a) Determine the (symmetric) matrix A corresponding to this quadratic form, i.e., Q(x) = x² Ax with x = [x1 x2 x3]. b) Diagonalize the matrix A in the form A = SAST, with S an orthogonal matrix containing (normalized) eigenvectors and A a diagonal matrix containing the eigenvalues. c) Using the elimination method, decompose the quadratic form Q(x) as a sum of squares of independent linear forms. What is the relationship between the signature of the quadratic form and the eigenvalues computed in b)? d) Using the results in b) or c), determine (including a short explanation!) a. the rank of the matrix A. b. if the matrix A is positive definite, negative definite, or indefinite. c. if the determinant of the matrix a is postive or negative.
Consider the quadratic form Q(x1, x2, x3) = 2x² + 2x² + 2x² + 2x1x2 + 2x1x3 + 2x2x3 a) Determine the (symmetric) matrix A corresponding to this quadratic form, i.e., Q(x) = x² Ax with x = [x1 x2 x3]. b) Diagonalize the matrix A in the form A = SAST, with S an orthogonal matrix containing (normalized) eigenvectors and A a diagonal matrix containing the eigenvalues. c) Using the elimination method, decompose the quadratic form Q(x) as a sum of squares of independent linear forms. What is the relationship between the signature of the quadratic form and the eigenvalues computed in b)? d) Using the results in b) or c), determine (including a short explanation!) a. the rank of the matrix A. b. if the matrix A is positive definite, negative definite, or indefinite. c. if the determinant of the matrix a is postive or negative.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter10: Matrices
Section10.CR: Chapter 10 Review
Problem 15CR
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![Consider the quadratic form Q(x1, x2, x3) = 2x² + 2x² + 2x² + 2x1x2 + 2x1x3 + 2x2x3
a) Determine the (symmetric) matrix A corresponding to this quadratic form, i.e., Q(x) = x² Ax
with x = [x1 x2 x3].
b) Diagonalize the matrix A in the form A = SAST, with S an orthogonal matrix containing
(normalized) eigenvectors and A a diagonal matrix containing the eigenvalues.
c) Using the elimination method, decompose the quadratic form Q(x) as a sum of squares of
independent linear forms. What is the relationship between the signature of the quadratic
form and the eigenvalues computed in b)?
d) Using the results in b) or c), determine (including a short explanation!)
a. the rank of the matrix A.
b. if the matrix A is positive definite, negative definite, or indefinite.
c. if the determinant of the matrix a is postive or negative.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6070c27-0824-4884-a918-195c8f609349%2F62f6db9c-5e8a-4d93-b67d-27ff378c1a54%2F2f6ow5b_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the quadratic form Q(x1, x2, x3) = 2x² + 2x² + 2x² + 2x1x2 + 2x1x3 + 2x2x3
a) Determine the (symmetric) matrix A corresponding to this quadratic form, i.e., Q(x) = x² Ax
with x = [x1 x2 x3].
b) Diagonalize the matrix A in the form A = SAST, with S an orthogonal matrix containing
(normalized) eigenvectors and A a diagonal matrix containing the eigenvalues.
c) Using the elimination method, decompose the quadratic form Q(x) as a sum of squares of
independent linear forms. What is the relationship between the signature of the quadratic
form and the eigenvalues computed in b)?
d) Using the results in b) or c), determine (including a short explanation!)
a. the rank of the matrix A.
b. if the matrix A is positive definite, negative definite, or indefinite.
c. if the determinant of the matrix a is postive or negative.
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