3 (a) Assuming that b ‡ 0, find an orthogonal matrix that diagonalises the matrix A = [%]. That is, find an orthogonal matrix ℗ such that PT AP is a diagonal matrix.
3 (a) Assuming that b ‡ 0, find an orthogonal matrix that diagonalises the matrix A = [%]. That is, find an orthogonal matrix ℗ such that PT AP is a diagonal matrix.
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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![3 (a) Assuming that b ‡ 0, find an orthogonal matrix that diagonalises the matrix A = [%]. That is, find
an orthogonal matrix P such that PT AP is a diagonal matrix.
(b) Determine the sign property of the quadratic form Q: R² → R given by Q(x, y) = x² + 4xy + y², by
(i) completing the square,
(ii)
using the eigenvalue test,
(iii) using Sylvester's criteria.
(c) Using a linear transformation x = PX, where P is an orthogonal matrix, x = [] and X = [],
express Q in diagonal form in the variables X and Y, and then in the original variables x and y.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc3c0612-e198-45c8-8e80-98d06c0b64b4%2Fa060628f-96e2-48f2-b176-7a2aae741919%2Fb7ct4sis_processed.png&w=3840&q=75)
Transcribed Image Text:3 (a) Assuming that b ‡ 0, find an orthogonal matrix that diagonalises the matrix A = [%]. That is, find
an orthogonal matrix P such that PT AP is a diagonal matrix.
(b) Determine the sign property of the quadratic form Q: R² → R given by Q(x, y) = x² + 4xy + y², by
(i) completing the square,
(ii)
using the eigenvalue test,
(iii) using Sylvester's criteria.
(c) Using a linear transformation x = PX, where P is an orthogonal matrix, x = [] and X = [],
express Q in diagonal form in the variables X and Y, and then in the original variables x and y.
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