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.