Skew cyclic displacements and decompositions of inverse matrix for an innovative structure matrix
-
2088
Downloads
-
4322
Views
Authors
Xiaoyu Jiang
- Department of Information and Telecommunications Engineering, The University of Suwon, Wau-ri, Bongdam-eup, Hwaseong-si, Gyeonggi-do, 445-743, Korea.
Kicheon Hong
- Department of Information and Telecommunications Engineering, The University of Suwon, Wau-ri, Bongdam-eup, Hwaseong-si, Gyeonggi-do, 445-743, Korea.
Zunwei Fu
- Department of Mathematics, The University of Suwon, Wau-ri, Bongdam-eup, Hwaseong-si, Gyeonggi-do, 445-743, Korea.
Abstract
In this paper, we study mainly on a class of column upper-minus-lower (CUML) Toeplitz matrices without standard Toeplitz structure, which are `` similar'' to the Toeplitz matrices. Their (-1,-1)-cyclic displacements coincide with cyclic displacement of some standard Toeplitz matrices. We obtain the formula on representation for the inverses of CUML Toeplitz matrices in the form of sums of products of (-1, 1)-circulants and (1, -1)-circulants factor by constructing the corresponding displacement of the matrices. In addition, based on the relation between CUML Toeplitz matrices and CUML Hankel matrices, the inverse formula of CUML Hankel matrices can also be obtained.
Share and Cite
ISRP Style
Xiaoyu Jiang, Kicheon Hong, Zunwei Fu, Skew cyclic displacements and decompositions of inverse matrix for an innovative structure matrix, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 8, 4058--4070
AMA Style
Jiang Xiaoyu, Hong Kicheon, Fu Zunwei, Skew cyclic displacements and decompositions of inverse matrix for an innovative structure matrix. J. Nonlinear Sci. Appl. (2017); 10(8):4058--4070
Chicago/Turabian Style
Jiang, Xiaoyu, Hong, Kicheon, Fu, Zunwei. "Skew cyclic displacements and decompositions of inverse matrix for an innovative structure matrix." Journal of Nonlinear Sciences and Applications, 10, no. 8 (2017): 4058--4070
Keywords
- CUML Toeplitz matrix
- CUML Hankel matrix
- skew cyclic displacement
- RSFPLR circulants
- RFMLR circulants
- decomposition
- inverse.
MSC
References
-
[1]
G. Ammar, P. Gader , New decompositions of the inverse of a Toeplitz matrix, Signal processing, scattering and operator theory, and numerical methods, Amsterdam, (1989), Progr. Systems Control Theory, Birkhäuser Boston, Boston, MA, 5 (1990), 421–428.
-
[2]
G. Ammar, P. Gader , A variant of the Gohberg-Semencul formula involving circulant matrices, SIAM J. Matrix Anal. Appl., 12 (1991), 534–540.
-
[3]
Y.-Q. Bai, T.-Z. Huang, X.-M. Gu , Circulant preconditioned iterations for fractional diffusion equations based on Hermitian and skew-Hermitian splittings, Appl. Math. Lett., 48 (2015), 14–22.
-
[4]
A. Ben-Artzi, T. Shalom , On inversion of Toeplitz and close to Toeplitz matrices, Linear Algebra Appl., 75 (1986), 173–192.
-
[5]
A. Brown, P. R. Halmos , Algebraic properties of Toeplitz operators, J. Reine Angew. Math., 213 (1964), 89–102.
-
[6]
C. C. Cowen , Hyponormality of Toeplitz operators , Proc. Amer. Math. Soc., 103 (1988), 809–812.
-
[7]
D. R. Farenick, W. Y. Lee, Hyponormality and spectra of Toeplitz operators, Trans. Amer. Math. Soc., 348 (1996), 4153–4174.
-
[8]
P. D. Gader , Displacement operator based decompositions of matrices using circulants or other group matrices, Linear Algebra Appl., 139 (1990), 111–131.
-
[9]
I. Gohberg, V. Olshevsky, Circulants, displacements and decompositions of matrices, Integral Equations Operator Theory, 15 (1992), 730–743.
-
[10]
G. Heinig , On the reconstruction of Toeplitz matrix inverses from columns, Linear Algebra Appl., 350 (2002), 199–212.
-
[11]
I. S. Hwang, W. Y. Lee , Block Toeplitz operators with rational symbols, J. Phys. A, 41 (2008), 7 pages.
-
[12]
Z.-L. Jiang, J.-X. Chen, The explicit inverse of nonsingular conjugate-Toeplitz and conjugate-Hankel matrices, J. Appl. Math. Comput., 53 (2017), 1–16.
-
[13]
Z.-L. Jiang, X.-T. Chen, J.-M. Wang, The explicit inverses of CUPL-Toeplitz and CUPL-Hankel matrices, East Asian J. Appl. Math., 7 (2017), 38–54.
-
[14]
X.-Y. Jiang, K.-C. Hong, Explicit determinants of the k-Fibonacci and k-Lucas RSFPLR circulant matrix in codes, Inf. Comput. Appl., Springer, Berlin, Heidelberg, (2013), 625–637.
-
[15]
X.-Y. Jiang, K.-C. Hong, Algorithms for finding inverse of two patterned matrices over \(Z_p\), Abstr. Appl. Anal., 2014 (2014), 6 pages.
-
[16]
X.-Y. Jiang, K.-C. Hong, , Exact determinants of some special circulant matrices involving four kinds of famous numbers, Abstr. Appl. Anal., 2014 (2014), 12 pages.
-
[17]
X.-Y. Jiang, K.-C. Hong , Explicit inverse matrices of Tribonacci skew circulant type matrices , Appl. Math. Comput., 268 (2015), 93–102.
-
[18]
X.-Y. Jiang, K.-C. Hong , Skew cyclic displacements and inversions of two innovative patterned matrices, Appl. Math. Comput., 308 (2017), 174–184.
-
[19]
Z. L. Jiang, Y. C. Qiao, S. D. Wang, Norm equalities and inequalities for three circulant operator matrices, Acta Math. Sin. (Engl. Ser.), 33 (2017), 571–590.
-
[20]
Z.-L. Jiang, T.-Y. Tam, Y.-F. Wang , Inversion of conjugate-Toeplitz matrices and conjugate-Hankel matrices, Linear Multilinear Algebra, 65 (2017), 256–268.
-
[21]
Z.-L. Jiang, D.-D. Wang, Explicit group inverse of an innovative patterned matrix , Appl. Math. Comput., 274 (2016), 220–228.
-
[22]
Z.-L. Jiang, T.-T. Xu , Norm estimates of \(\omega\)-circulant operator matrices and isomorphic operators for \(\omega\)-circulant algebra, Sci. China Math., 59 (2016), 351–366.
-
[23]
T. Kailath, S. Y. Kung, M. Morf , Displacement ranks of matrices and linear equations, J. Math. Anal. Appl., 68 (1979), 395–407.
-
[24]
S.-L. Kong, Z.-S. Zhang , Optimal control of stochastic system with Markovian jumping and multiplicative noises, Acta Automat. Sinica, 38 (2017), 1113–1118.
-
[25]
G. Labahn, T. Shalom, Inversion of Toeplitz matrices with only two standard equations, Linear Algebra Appl., 175 (1992), 143–158.
-
[26]
G. Labahn, T. Shalom, Inversion of Toeplitz structured matrices using only standard equations , Linear Algebra Appl., 207 (1994), 49–70.
-
[27]
L. Lerer, M. Tismenetsky , Generalized Bezoutian and the inversion problem for block matrices, I , General scheme, Integral Equations Operator Theory, 9 (1986), 790–819.
-
[28]
X.-G. Lv, T.-Z. Huang, A note on inversion of Toeplitz matrices, Appl. Math. Lett., 20 (2007), 1189–1193.
-
[29]
M. K. Ng, J.-Y. Pan, Weighted Toeplitz regularized least squares computation for image restoration, SIAM J. Sci. Comput., 36 (2014), B94–B121.
-
[30]
M. K. Ng, K. Rost, Y.-W. Wen , On inversion of Toeplitz matrices, Linear Algebra Appl., 348 (2002), 145–151.
-
[31]
D. Potts, G. Steidl , Preconditioners for ill-conditioned Toeplitz matrices, BIT, 39 (1999), 513–533.
-
[32]
N. Shen, Z.-L. Jiang, J. Li, On explicit determinants of the RFMLR and RLMFL circulant matrices involving certain famous numbers , WSEAS Trans. Math., 12 (2013), 42–53.
-
[33]
Y.-P. Zheng, S.-G. Shon , Exact determinants and inverses of generalized Lucas skew circulant type matrices , Appl. Math. Comput., 270 (2015), 105–113.
-
[34]
Y.-P. Zheng, S.-G. Shon, J.-Y. Kim, Cyclic displacements and decompositions of inverse matrices for CUPL Toeplitz matrices, J. Math. Anal. Appl., 455 (2017), 727–741.
