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Mathematics of Computation

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The generalized triangular decomposition


Authors: Yi Jiang, William W. Hager and Jian Li
Journal: Math. Comp. 77 (2008), 1037-1056
MSC (2000): Primary 15A23, 65F25, 94A11, 60G35
DOI: https://doi.org/10.1090/S0025-5718-07-02014-5
Published electronically: October 1, 2007
MathSciNet review: 2373191
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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:

$\displaystyle \prod_{i=1}^k \vert r_{i}\vert \le \prod_{i=1}^k \sigma_i, \; \; 1 \le k < K, \quad \prod_{i=1}^K \vert r_{i}\vert = \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.


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Additional Information

Yi Jiang
Affiliation: Department of Electrical and Computer Engineering, University of Florida, P.O. Box 116130, Gainesville, Florida 32611-6130
Address at time of publication: Department of Electrical and Computer Engineering, University of Colorado, Boulder, Colorado 80309-0425
Email: yjiang@dsp.ufl.edu

William W. Hager
Affiliation: Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105
Email: hager@math.ufl.edu

Jian Li
Affiliation: Department of Electrical and Computer Engineering, P.O. Box 116130, University of Florida, Gainesville, Florida 32611-6130
Email: li@dsp.ufl.edu

DOI: https://doi.org/10.1090/S0025-5718-07-02014-5
Keywords: Generalized triangular decomposition, geometric mean decomposition, matrix factorization, unitary factorization, singular value decomposition, Schur decomposition, MIMO systems, inverse eigenvalue problems
Received by editor(s): July 21, 2005
Received by editor(s) in revised form: June 15, 2006
Published electronically: October 1, 2007
Additional Notes: This material is based on work supported by the National Science Foundation under Grants 0203270, 0619080, 0620286, and CCR-0097114.
Article copyright: © Copyright 2007 American Mathematical Society