## Inverses and determinants of Toeplitz-Hessenberg matrices.(English)Zbl 1403.15021

Summary: The inverses of Toeplitz-Hessenberg matrices are investigated. It is known that each inverse of such a matrix is a sum of a lower triangular matrix $$L$$ and a matrix $$R$$ of rank 1. The formulas of $$L$$ and $$x$$, $$y$$ such that $$xy^T = R$$ are derived. Using this result we propose an algorithm for inverting Toeplitz-Hessenberg matrices. Moreover, from the expression of the inverse a formula for the determinant is deduced.

### MSC:

 15B05 Toeplitz, Cauchy, and related matrices 15A09 Theory of matrix inversion and generalized inverses 15A15 Determinants, permanents, traces, other special matrix functions 65F05 Direct numerical methods for linear systems and matrix inversion

### Keywords:

Hessenberg matrix; Toeplitz matrix; matrix inverse; determinant
Full Text:

### References:

 [1] J. Abderramán Marrero and V. Tomeo, On the closed representation for the inverses of Hessenberg matrices, J. Comput. Appl. Math. 236 (2012), no. 12, 2962–2970. · Zbl 1237.15005 [2] J. Abderramán Marrero, V. Tomeo and E. Torrano, On inverses of infinite Hessenberg matrices, J. Comput. Appl. Math. 275 (2015), 356–365. · Zbl 1297.15005 [3] E. Asplund, Inverses of matrices $$\{a_{ij}\}$$ which satisfy $$a_{ij} = 0$$ for $$j > i+p$$, Math. Scand. 7 (1959), 57–60. · Zbl 0093.24102 [4] B. Bukhberger and G. A. Emel’yanenko, Methods of inverting tridiagonal matrices, USSR Comput. Math. and Math. Phys. 13 (1973), no. 3, 10–20. · Zbl 0281.65023 [5] D. K. Faddeev, Some properties of a matrix that is the inverse of a Hessenberg matrix, Numerical Methods and Questions in the Organization of Calculations 5, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 111 (1981), 177–179. · Zbl 0473.15011 [6] Y. Ikebe, On inverses of Hessenberg matrices, Linear Algebra Appl. 24 (1979), 93–97. · Zbl 0397.15005 [7] J. Maroulas, Factorization of Hessenberg matrices, Linear Algebra Appl. 506 (2016), 226–243. · Zbl 1346.15014 [8] M. Merca, A note on the determinant of a Toeplitz-Hessenberg matrix, Spec. Matrices 2 (2014), 10–16. · Zbl 1291.15015 [9] T. Muir, A Treatise on the Theory of Determinants, Revised and enlarged by William H. Metzler, Dover, New York, 1960. [10] M. J. Piff, Inverses of banded and $$k$$-Hessenberg matrices, Linear Algebra Appl. 85 (1987), 9–15. · Zbl 0607.15004 [11] L. Verde-Star, Elementary triangular matrices and inverses of $$k$$-Hessenberg and triangular matrices, Spec. Matrices 3 (2015), 250–256. · Zbl 1329.15073 [12] X. Zhong, On inverses and generalized inverses of Hessenberg matrices, Linear Algebra Appl. 101 (1988), 167–180. · Zbl 0651.65030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.