Diagonalising symmetric matrices and affine transformations.

I have just finished the last book in the linear algebra block of the course and I will be starting the first analysis block next. Before I do I just want to comment on a couple of things that came up in LA5.

It is stated a couple of times (see page 31 and page 33) that a symmetric nxn matrix always has an orthonormal eigenvector basis, so that it will always be orthogonally diagonalisable. I don't understand why this is not stated properly as a theorem and it makes me wonder if I have missed something. On page 33 Theorem 3.1 states that Eigenvectors corresponding to distinct eigenvalues of a symmetric matrix are orthogonal. Proof of this theorem is given on page 35 but this only goes some way to explain why a symmetric matrix always has an orthonormal eigenvector basis. What happens if the eigenvalues of a symmetric matrix are not distinct?

I actually think that this theorem may be very important because in physics, for example, symmetries play an important part in conservation laws. I wish this had been dealt with a bit better as it seems like a big result.

The other thing that is puzzling me appears in section 4 on conics and quadrics. Strategy 4.1 on page 43 uses matrices to transform the coordinates of a conic into a different system where the equation of the conic is recognised as being in 'standard form'. What puzzles me is that we are supposed to be dealing with linear transformations and yet part of the strategy involves translating the origin. If the conic is not centred at the origin, then surely we have to use affine transformations (which do not preserve the origin) rather than linear transformations (which do). It all looks rather like a trick because here we are using linear transformations to solve an affine transformation problem. I guess we aren't seeing the full details.