. Apply orthogonal diagonalisa - tion to the symmetric matrix S = XXXXXXXXXX(a) Find the eigenvalues...

Question

. Apply orthogonal diagonalisa - tion to the symmetric matrix S = XXXXXXXXXX
(a) Find the eigenvalues of S . (b) One of the eigenvalues has multiplicity equal to two. Find two orthogonal eigenvectors for this eigenvalue, and find an eigenvector for the second eigenvalue. (c) Verify that the last eigenvector is orthogonal to the two previous ones. (d) Let M be the 3 x 3 matrix whose columns are the eigenvectors you have found. Evaluate /117./11, with as little computation as possible. Give reasons for any computations you were able to omit. (e) Construct. three pairwise orthogonal eigenvectors of unit length, and find au orthogonal matrix V such that S= V D 17-' with D a diagonal matrix.
2. Simplify the expression I XXXXXXXXXXei0 I.
.r.' + 2x5 + x4 + 16x3 + 2.t XXXXXXXXXXFactorise the denominator of f (s) completely, and use this to write f (x) 2x4 373 -x2 - 2x -- 1 as a partial fraction. Use the result to find all anti-derivatives of ,f (x) .
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Robert · Accepted Answer

Problem 1: 
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c)  
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Therefore, the two vectors are orthogonal.

. Apply orthogonal diagonalisa - tion to the symmetric matrix S = XXXXXXXXXX (a) Find the eigenvalues of S . (b) One of the eigenvalues has multiplicity equal to two. Find two orthogonal eigenvectors...

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