The generalized triangular decomposition

Abstract

Given a complex matrix $\mathbf {H}$, we consider the decomposition $\mathbf {H} = \mathbf {QRP}^*$, where $\mathbf {R}$ is upper triangular and $\mathbf {Q}$ and $\mathbf {P}$ have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where $\mathbf {R}$ is an upper triangular matrix with the eigenvalues of $\mathbf {H}$ on the diagonal. We show that any diagonal for $\mathbf {R}$ can be achieved that satisfies Weyl’s multiplicative majorization conditions: \[ \prod _{i=1}^k |r_{i}| \le \prod _{i=1}^k \sigma _i, \; \; 1 \le k < K, \quad \prod _{i=1}^K |r_{i}| = \prod _{i=1}^K \sigma _i, \] where $K$ is the rank of $\mathbf {H}$, $\sigma _i$ is the $i$-th largest singular value of $\mathbf {H}$, and $r_{i}$ is the $i$-th largest (in magnitude) diagonal element of $\mathbf {R}$. Given a vector $\mathbf {r}$ which satisfies Weyl’s conditions, we call the decomposition $\mathbf {H} = \mathbf {QRP}^*$, where $\mathbf {R}$ is upper triangular with prescribed diagonal $\mathbf {r}$, the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.