Skip to main content
Log in

Compound matrices: properties, numerical issues and analytical computations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aitken, A.C.: Determinants and Matrices. Oliver & Boyd, Edinburgh (1967)

    Google Scholar 

  2. Bernstein, D.S.: Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press, Princeton (2005)

    MATH  Google Scholar 

  3. 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)

  4. 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)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fiedler, M.: Special Matrices and Their Applications in Numerical Mathematics. Martinus Nijhoff, Dordrecht (1986)

    MATH  Google Scholar 

  6. Hadamard, J.: Résolution d’une question relative aux déterminants. Bull. Sci. Math. 17, 240–246 (1893)

    Google Scholar 

  7. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  8. 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)

    Article  MATH  MathSciNet  Google Scholar 

  9. 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)

    Article  MATH  MathSciNet  Google Scholar 

  10. Karcanias, N., Laios, B., Giannakopoulos, C.: Decentralized determinantal assignment problem: fixed and almost fixed modes and zeros. Internat. J. Control 48, 129–147 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  11. 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)

    MATH  MathSciNet  Google Scholar 

  12. Kravvaritis, C., Mitrouli, M.: An algorithm to find values of minors of Hadamard matrices. Bull. Greek Math. Soc. 54, 221–238 (2007)

    MATH  MathSciNet  Google Scholar 

  13. Marcus, M.: Finite Dimensional Multilinear Algebra, Two Volumes. Marcel Dekker, New York (1973–1975)

    Google Scholar 

  14. Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Boston (1964)

    MATH  Google Scholar 

  15. Mitrouli, M., Karcanias, N.: Computation of the GCD of polynomials using Gaussian transformation and shifting. Internat. J. Control 58, 211–228 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. 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)

    Google Scholar 

  17. Mitrouli, M., Karcanias, N., Koukouvinos, C.: Numerical aspects for nongeneric computations in control problems and related applications. Congr. Numer. 126, 5–19 (1997)

    MATH  MathSciNet  Google Scholar 

  18. Mitrouli, M., Koukouvinos, C.: On the computation of the Smith normal form of compound matrices. Numer. Algorithms 16, 95–105 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Nambiar, K.K., Sreevalsan, S.: Compound matrices and three celebrated theorems. Math. Comput. Model. 34, 251–255 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. 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)

    Article  MATH  MathSciNet  Google Scholar 

  21. 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)

    Article  MATH  Google Scholar 

  22. Zhang, F.: A majorization conjecture for Hadamard products and compound matrices. Linear Multilinear Algebra 33, 301–303 (1993)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marilena Mitrouli.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-008-9222-7

Keywords

Mathematics Subject Classifications (2000)

Navigation