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.

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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.
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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