Maple double check of problem #9 in 8.3

> with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

> A:=matrix([[-3,1,2],[1,-3,2],[2,2,0]]);

> eigenvalues(A);

> eigenvectors(A);

> GramSchmidt({vector([-1,1,0]),vector([-2,0,1])});

These are different than ours. Notice Maple used the second first as the first one in the G-S process. We switch them below.

> GramSchmidt({vector([-2,0,1]),vector([-1,1,0])});

We normalize the three eigenvectors, the two for eigenvalue -4 and the one for eigenvalue 2.

> normalize(vector([1,1,2 ]));
normalize(vector([-1,1,0]));
normalize(vector([-1,-1,1]));

> P:=transpose(matrix([[1/6*sqrt(6), 1/6*sqrt(6), 1/3*sqrt(6)],
[-1/2*sqrt(2), 1/2*sqrt(2), 0],[-1/3*sqrt(3), -1/3*sqrt(3), 1/3*sqrt(3)]]));

This is the same as the one we got by hand. Is in really orthogonal? Does in diagonalize A as the theory says it should?

> inverse(P);

Indeed P inverse equals P transpose.

> evalm(transpose(P)&*A&*P);

Check!

We use the jordon command to diagonialize A and construct P.

>

> jordan(A,'P');

> evalm(P);

jordan diagonalized A differently, and did not normalize P, nor is its P orthogonal.