Thread: Diagonalizing Symmetric Bilinear Forms

Find a 2x2 orthogonal matrix such that is diagonal where



So here's my working...

The eigenvalues of S are 10 and -5.
Thus the eigenvectors are and .

Now i know these are supposed to be scaled by which would surely give me,


But the answer has it as all of them are divided by ... Why is that? Do you scale all of the entries by the same eigenvalues and just choose any eigenvalue to use?

Also, am i right in saying S is a type (1,1) matrix?
And is this because there is a +ve and a -ve eigenvalue, or is it because you compute the determinant of the 1x1 matrix (i.e, the number 7), and the determinant of the 2x2 matrix (i.e, S) and since one is +ve and one is -ve that means its type (1,1)?

The scaling factor has nothing to do with the eigenvalue. It is the length of the eigenvector, which for both of these eigenvectors is .