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.
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 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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