The generalized triangular decomposition

Given a complex matrix H H , we consider the decomposition H = Q R P ∗ H = QRP^* , where R R is upper triangular and Q Q and P P have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where R R is an upper triangular matrix with the eigenvalues of H H on the diagonal. We show that any diagonal for R R can be achieved that satisfies Weyl’s multiplicative majorization conditions: \[ ∏ i = 1 k | r i | ≤ ∏ i = 1 k σ i , 1 ≤ k > K , ∏ i = 1