I presume, since you thought to find eigenvalues and eigenvectors, that you know that any symmetric matrix (and the matrix representing a quadratic is always symmetric) can be diagonalized: , where D is the diagonal matrix having the eigenvalues on the diagonal and P is a matrix having the eigenvalues as columns. If you choose the eigenvectors to be orthonormal (i.e. of unit length- they will necessarily be perpendicular), then P is the "rotation" matrix; it represents the rotation to take the xy-axes to the principal axes of the conic section defined by the quadratic.