The generalized triangular decomposition

被引:0
作者
Jiang, Yi [1 ]
Hager, William W. [2 ]
Li, Jian [1 ]
机构
[1] Univ Florida, Dept Elect & Comp Engn, Gainesville, FL 32611 USA
[2] Univ Florida, Dept Math, Gainesville, FL 32611 USA
关键词
generalized triangular decomposition; geometric mean decomposition; matrix factorization; unitary factorization; singular value decomposition; Schur decomposition; MIMO systems; inverse eigenvalue problems;
D O I
暂无
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Given a complex matrix H, we consider the decomposition H = QRP*, where R is upper triangular and Q and P have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where R is an upper triangular matrix with the eigenvalues of H on the diagonal. We show that any diagonal for R can be achieved that satisfies Weyl's multiplicative majorization conditions: (k)Pi(i =1) |r(i)| <= (k)Pi(i =1) sigma(i), 1 <= k < K, (k)Pi(i =1) |r(i)| = (k)Pi(i =1) sigma(i), where K is the rank of H, sigma(i) is the i-th largest singular value of H, and r(i) is the i-th largest (in magnitude) diagonal element of R. Given a vector r which satisfies Weyl's conditions, we call the decomposition H = QRP*, where R is upper triangular with prescribed diagonal 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.
引用
收藏
页码:1037 / 1056
页数:20
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