[east2west]

I think that sentence is confusing but mathematical conclusion still stands. For quadratic forms, we are only concerned with symmetric matrices, or the symmetric part of any real matrix, because we can always replace any square matrix in a quadratic form, x^T A x, by x^T (A + A^T)/2 x, the symmetric part of the matrix, and the quadratic form still retains the original value. Then, for symmetric matrices, their eigenvalues being positive implies that the matrices are positive definite. Symmetric square matrices have real eigenvalues; one can prove this by taking the complement of quadratic forms.