Since if
implies
is symmetric, there exists an orthogonal matrix
such that
and
is a diagonal matrix with the eigenvalues of
along the diagonal and the columns of
are the eigenvectors of
.
is called a unitary transformation. Since
is orthogonal,
and thus
Further, since
when
is nonsingular,
is positive definite and thus has positive eigenvalues. Thus
where
has the square roots of the eigenvalues of
on the diagonal.
Then
where
. So the problem is reduced to finding the eigenvalues and eigenvectors of a symmetric matrix
There are efficient numerical methods for this.