Elementary triangular matrices and inverses of k-Hessenberg and triangular matrices

Luis Verde-Star

Special Matrices (2015)

  • Volume: 3, Issue: 1, page 250-256, electronic only
  • ISSN: 2300-7451

Abstract

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We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its nonzero entries in the k-th column,where 1 ≤ k ≤ n.

How to cite

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Luis Verde-Star. "Elementary triangular matrices and inverses of k-Hessenberg and triangular matrices." Special Matrices 3.1 (2015): 250-256, electronic only. <http://eudml.org/doc/275834>.

@article{LuisVerde2015,
abstract = {We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its nonzero entries in the k-th column,where 1 ≤ k ≤ n.},
author = {Luis Verde-Star},
journal = {Special Matrices},
keywords = {Triangular matrices; factorization; k-Hessenberg matrices; matrix inversion; triangular matrices; -Hessenberg matrices},
language = {eng},
number = {1},
pages = {250-256, electronic only},
title = {Elementary triangular matrices and inverses of k-Hessenberg and triangular matrices},
url = {http://eudml.org/doc/275834},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Luis Verde-Star
TI - Elementary triangular matrices and inverses of k-Hessenberg and triangular matrices
JO - Special Matrices
PY - 2015
VL - 3
IS - 1
SP - 250
EP - 256, electronic only
AB - We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its nonzero entries in the k-th column,where 1 ≤ k ≤ n.
LA - eng
KW - Triangular matrices; factorization; k-Hessenberg matrices; matrix inversion; triangular matrices; -Hessenberg matrices
UR - http://eudml.org/doc/275834
ER -

References

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  1. [1] M. Elouafi, A. Driss Aiat Hadj, A new recursive algorithm for inverting Hessenberg matrices, Appl. Math. Comp., 214 (2009) 497–499. Zbl1170.65311
  2. [2] Y. Ikebe, On inverses of Hessenberg matrices, Linear Algebra Appl., 24 (1979) 93–97. Zbl0397.15005
  3. [3] M. J. Piff, Inverses of banded and k-Hessenberg matrices, Linear Algebra Appl., 85 (1987) 9–15. Zbl0607.15004
  4. [4] W. Shur, A simple closed form for triangular matrix powers, Electr. J. Linear Algebra, 22 (2011) 1000–1003. Zbl1254.15031
  5. [5] L. Verde-Star, Infinite triangular matrices, q-Pascal matrices, and determinantal representations, Linear Algebra Appl., 434 (2011) 307–318. Zbl1203.15021
  6. [6] L. Verde-Star, Divided differences and linearly recurrent sequences, Stud. Appl. Math. 95 (1995) 433–456. Zbl0843.65094
  7. [7] L. Verde-Star, Functions of matrices, Linear Algebra Appl., 406 (2005) 285–300. 
  8. [8] Z. Xu, On inverses and generalized inverses of Hessenberg matrices, Linear Algebra Appl., 101 (1988) 167–180. Zbl0651.65030
  9. [9] T. Yamamoto, Y. Ikebe, Inversion of band matrices, Linear Algebra Appl., 24 (1979) 105–111. Zbl0408.15004

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