I'm just trying to clear a few things up about diagonalising a matrix. As I understand it, if you have a Euclidean space and a symmetric matrix, there is essentially two ways of diagonalising this matrix.
One is for the quadratic form generated by this matrix, finding an orthogonal basis which makes the matrix for this quadratic form a diagonal. This is the same as finding an invertible matrix $P$ such that $P^T\!\!AP$ is diagonal.
The second is for the linear operator generated by this matrix, finding an orthonormal basis consisting of eigenvectors of T. Which is the same as finding an orthogonal matrix $P$ (which consist of the normalised eigenvectors) such that $P^T \!\! AP$ is diagonal (which consists of the eigenvalues of $V$).
Have I got this right? So for a symmetric matrix on a Euclidean space there is two ways of diagonalising the matrix?