Reducing a symmetric matrix to diagonal form

Does anyone know what the point of this process actually is? I have no problem doing it but the only relevence to the original matrix seems to be its eigenvalues on the diagonal. I have no doubt that there is some reason for doing this and I would appreciate if someone could expain it to me, thanks.

This is one of the reasons why I dislike the treatment of linear algebra at A-level. In fact diagonalisable matrices are the nicest matrices there are, and it's a big key fact that symmetric matrices are diagonalisable. It's useful in many proofs because it means there is a basis of eigenvectors for the matrix. So many proofs in linear algebra would be much simpler if we could assume matrices were diagonalisable, and in fact a lot of work goes into proving that there is a next-best thing (Jordan normal form - all matrices can be put into a form that is "nearly diagonal").

Essentially, it's really really useful to know that it is possible to diagonalise symmetric matrices, but at A-level there are no good reasons for you to do it. (Much like the A-level treatment of groups.)

Thanks for your answer. I suppose I can understand the logic of including this in the syllabus if it becomes very useful later on but I think this is the wrong attitude on the part of the exam board. I think it would be a lot more sensible to study things in more detail rather than cramming loads of topics in to an exam and only looking at them superficially.