2007/07/12 by Brian Sutton, Sutton, Brian D. · 2 citations
Computer Science · Engineering · Mathematics · #15A18 #15A23 #65F15 #Algorithm #Block (permutation group theory) #Block matrix #Combinatorics #Computation #Computer science #Decomposition #FOS: Mathematics #Mathematics #Matrix (chemical analysis) #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Optical Network Technologies #Physics #Singular value #Singular value decomposition #Tensor decomposition and applications #Unitary matrix #Unitary state
paper · pdf · doi:10.48550/arxiv.0707.1838
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2007/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
An algorithm is developed to compute the complete CS decomposition (CSD) of a partitioned unitary matrix. Although the existence of the CSD has been recognized since 1977, prior algorithms compute only a reduced version (the 2-by-1 CSD) that is equivalent to two simultaneous singular value decompositions. The algorithm presented here computes the complete 2-by-2 CSD, which requires the simultaneous diagonalization of all four blocks of a unitary matrix partitioned into a 2-by-2 block structure. The algorithm appears to be the only fully specified algorithm available. The computation occurs in two phases. In the first phase, the unitary matrix is reduced to bidiagonal block form, as described by Sutton and Edelman. In the second phase, the blocks are simultaneously diagonalized using techniques from bidiagonal SVD algorithms of Golub, Kahan, and Demmel. The algorithm has a number of desirable numerical features.