Eigenvalues of the unitary part of a matrix
Alfred Horn, Robert Steinberg · Pacific Journal of Mathematics · 1959
LEMMA 3. If (X t ), (a t ) is realizable and z is any complex number of modulus 1, then (z\) f (za) is realizable.LEMMA 4. If (μ 19 μ 2 ) results from (\ 19 λ 2 ) by a pinch and T is a triangular matrix with diagonals elements X lf λ 2 , then T is congruent to a matrix with eigenvalues μ 19 μ 2 .Proof.By multiplication by a suitable constant, we may suppose