Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

A MATRIX PENCIL APPROACH COMPUTING THE ELEMENTARY DIVISORS OF A MATRIX : NUMERICAL ASPECTS AND APPLICATIONS

  • Mitrouli, M. (Department of Mathematics University of Athens) ;
  • Kalogeropoulos, G. (Department of Mathematics University of Athens)
  • Published : 1998.09.01

⟨ Previous Next ⟩

Abstract

In the present paper is presented a new matrix pencil-based numerical approach achieving the computation of the elemen-tary divisors of a given matrix $A \in C^{n\timesn}$ This computation is at-tained without performing similarity transformations and the whole procedure is based on the construction of the Piecewise Arithmetic Progression Sequence(PAPS) of the associated pencil $\lambda I_n$ -A of matrix A for all the appropriate values of $\lambda$ belonging to the set of eigenvalues of A. This technique produces a stable and accurate numerical algorithm working satisfactorily for matrices with a well defined eigenstructure. The whole technique can be applied for the computation of the first second and Jordan canonical form of a given matrix $A \in C^{n\timesn}$. The results are accurate for matrices possessing a well defined canonical form. In case of defective matrices indications of the most appropriately computed canonical form. In case of defective matrices indication of the most appropriately computed canonical form are given.

Keywords

References

  1. Holt-Saunders International Editions Linear System Theory and Design Chen, C.T.
  2. ACM Trans. Math. Soft v.19 The generalized Schur decomposition of an arbitrary pencil A―$\lambda$ B : robust software with error bounds and applications Demmel,J.;Kagstrom,B.
  3. Lin. Alg. and its Appl. v.74 Rank and Null space calculations using matrix decomposition without column interchanges Foster,L.
  4. Theory of Matrics, 1,2, Chelsea Gantmacher,G.
  5. Matrix Computations Golub,G.;Van Loan,C
  6. Dept. of Comp. Scien Rank Degeneracy and Least Squares Problems, STAN-CS-76-559 Golub,G;Klema,V.;Stewart,G.
  7. SIAM Review v.18 no.4 Ill-Conditioned Eigensystems and the Computation of the Jordan Canonical Form Golub,G.;Wilkinson,J.
  8. ACM Transactions on Math. Software v.6 no.3 An Algorithm for Numerical Computation of the Jardan Normal Form of a Complex Matrix Kagstrom,B.;Ruhe,A.
  9. Matrix Pencils and Linear Systems Theory, PH. D. Thesis Kalogeropoulos,G.
  10. Control and Computers v.19 On the Computation of the Weierstrass Canonical form of a regular matrix pencil Kalogeropoulos,G.;Mitrouli.M,
  11. Control and Computers On the Computation of the Weierstrass Canonical form of a regular matrix pencil: PartⅡ Kalogeropoulos,G.;Mitrouli,M.
  12. Jour. Inst. Math. and Comp. Sci.(Math. Ser.) v.7 On the Computation of column and row minimal induces of a singular pencil Kalogeropoulos,G.;Mitrouli,M.
  13. Theory of Matrices Lancaster,P.
  14. Numerical Algorithms v.7 A Compound-Matrix algorithm for the computation of the Smith form of a polynomial matrix Mitrouli,M.;Kalogeropoulos,G.
  15. An algorithm for numerical determination of the structure of a general matrix v.10 Ruhe,A.
  16. Lin. Alg. Appl. v.27 The computation of Kronecker's canonical form of a singular pencil Van Dooren,P.
  17. The Algebraic Eigenvalue Problem Wilkinson,J.