Abstract
This paper studies the possibility to calculate efficiently compounds of real matrices which have a special form or structure. The usefulness of such an effort lies in the fact that the computation of compound matrices, which is generally noneffective due to its high complexity, is encountered in several applications. A new approach for computing the Singular Value Decompositions (SVD’s) of the compounds of a matrix is proposed by establishing the equality (up to a permutation) between the compounds of the SVD of a matrix and the SVD’s of the compounds of the matrix. The superiority of the new idea over the standard method is demonstrated. Similar approaches with some limitations can be adopted for other matrix factorizations, too. Furthermore, formulas for the n − 1 compounds of Hadamard matrices are derived, which dodge the strenuous computations of the respective numerous large determinants. Finally, a combinatorial counting technique for finding the compounds of diagonal matrices is illustrated.
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References
Aitken, A.C.: Determinants and Matrices. Oliver & Boyd, Edinburgh (1967)
Bernstein, D.S.: Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press, Princeton (2005)
Boutin, D.L., Gleeson R.F., Williams, R.M.: Wedge Theory / Compound Matrices: Properties and Applications. Office of Naval Research, Arlington, Report number NAWCADPAX–96-220-TR. http://handle.dtic.mil/100.2/ADA320264 (1996)
Elsner, L., Hershkowitz, D., Schneider, D.: Bounds on norms of compound matrices and on products of eigenvalues. Bull. Lond. Math. Soc. 32, 15–24 (2000)
Fiedler, M.: Special Matrices and Their Applications in Numerical Mathematics. Martinus Nijhoff, Dordrecht (1986)
Hadamard, J.: Résolution d’une question relative aux déterminants. Bull. Sci. Math. 17, 240–246 (1893)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Kaltofen, E., Krishnamoorthy, M.S., Saunders, B.D: Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM J. Algebr. Discrete Methods 8, 683–690 (1987)
Karcanias, N., Giannakopoulos, C.: Grassmann invariants, almost zeros and the determinantal zero pole assignment problems of linear systems. Internat. J. Control 40, 673–698 (1984)
Karcanias, N., Laios, B., Giannakopoulos, C.: Decentralized determinantal assignment problem: fixed and almost fixed modes and zeros. Internat. J. Control 48, 129–147 (1988)
Koukouvinos, C., Mitrouli, M., Seberry, J.: Numerical algorithms for the computation of the Smith normal form of integral matrices. Congr. Numer. 133, 127–162 (1998)
Kravvaritis, C., Mitrouli, M.: An algorithm to find values of minors of Hadamard matrices. Bull. Greek Math. Soc. 54, 221–238 (2007)
Marcus, M.: Finite Dimensional Multilinear Algebra, Two Volumes. Marcel Dekker, New York (1973–1975)
Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Boston (1964)
Mitrouli, M., Karcanias, N.: Computation of the GCD of polynomials using Gaussian transformation and shifting. Internat. J. Control 58, 211–228 (1993)
Mitrouli, M., Karcanias, N., Giannakopoulos, C.: The computational framework of the determinantal assignment problem. In: Proceedings ECC ’91 European Control Conference, vol. 1, pp. 98–103. ECC, Grenoble (1991)
Mitrouli, M., Karcanias, N., Koukouvinos, C.: Numerical aspects for nongeneric computations in control problems and related applications. Congr. Numer. 126, 5–19 (1997)
Mitrouli, M., Koukouvinos, C.: On the computation of the Smith normal form of compound matrices. Numer. Algorithms 16, 95–105 (1997)
Nambiar, K.K., Sreevalsan, S.: Compound matrices and three celebrated theorems. Math. Comput. Model. 34, 251–255 (2001)
Prells, U., Friswell, M.I., Garvey, S.D., Use of geometric algebra: compound matrices and the determinant of the sum of two matrices. Proc. R. Soc. Lond. A 459, 273–285 (2003)
Tsatsomeros, M., Maybee, J.S., Olesky, D.D., Driessche, P.V.D.: Nullspaces of matrices and their compounds. Linear Multilinear Algebra 34, 291–300 (1993)
Zhang, F.: A majorization conjecture for Hadamard products and compound matrices. Linear Multilinear Algebra 33, 301–303 (1993)
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Kravvaritis, C., Mitrouli, M. Compound matrices: properties, numerical issues and analytical computations. Numer Algor 50, 155–177 (2009). https://doi.org/10.1007/s11075-008-9222-7
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DOI: https://doi.org/10.1007/s11075-008-9222-7