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A numerical algorithm for block-diagonal decomposition of matrix \(*\)-algebras with general irreducible components. (English) Zbl 1204.65035

Summary: An algorithm is proposed for finding the finest simultaneous block-diagonalization of a finite number of square matrices, or equivalently the irreducible decomposition of a matrix \(*\)-algebra given in terms of its generators. This extends the approach initiated by Murota-Kanno-Kojima-Kojima. The algorithm, composed of numerical-linear algebraic computations, does not require any algebraic structure to be known in advance. The main ingredient of the algorithm is the Schur decomposition and its skew-Hamiltonian variant for eigenvalue computation.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A23 Factorization of matrices
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[1] Benner P., Kressner D.: Algorithm 854: Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices II. ACM Trans. Math. Softw. 32, 352–373 (2006) · Zbl 1365.65091 · doi:10.1145/1141885.1141895
[2] Benner, P., Kressner, D., Mehrmann, V.: Skew-Hamiltonian and Hamiltonian eigenvalue problems: Theory, algorithms and applications. In: Drmač, Z., Marušić, M., Tutek, Z. (eds.) Proc. Conf. Appl. Math. Sci. Comput., pp. 3–39. Springer, Brijoni, Croatia (2005) · Zbl 1069.65034
[3] Bunse-Gerstner A., Byers R., Mehrmann V.: A quaternion QR algorithm. Numer. Math. 55, 83–95 (1989) · Zbl 0681.65024 · doi:10.1007/BF01395873
[4] Murota, K., Kanno, Y., Kojima, M., Kojima, S.: A numerical algorithm for block-diagonal decomposition of matrix *-algebras with application to semidefinite programming. Jpn. J. Ind. Appl. Math. 27, (2010, in press) (Original version: ”A numerical algorithm for block-diagonal decomposition of matrix *-algebras,” METR 2007-52, University of Tokyo, September 2007) · Zbl 1204.65068
[5] Narayanan S.: Space Structures: Principles and Practice. Multi-Science Publishing, UK (2008)
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