The Partial Non-combinatorially Symmetric N10 -Matrix Completion Problem
An n×n matrix is called an N1 0 -matrix if all principal minors are non-positive and each entry is non-positive. In this paper, we study the partial non-combinatorially symmetric N1 0 -matrix completion problems if the graph of its specified entries is a transitive tournament or a double cycle. In general, these digraphs do not have N1 0 -completion. Therefore, we have given sufficient conditions that guarantee the existence of the N1 0 -completion for these digraphs.
[1] Gu-Fang Mou, Ting-Zhu Huang, The N10 -matrix completion problem, to
appear.
[2] T. Parthasathy, G. Ravindran, N-matrices, Linear Algebra Appl., 139
(1990) 89-102.
[3] Sheng-Wei Zhou, Ting-Zhu Huang, On Perron complements of inverse
N0-matrices, Linear Algebra Appl., 434 (2011) 2081-2088.
[4] L. DeAlba and L. Hogben, Completion problems of P-matrix patterns,
Linear Algebra Appl., 319 (2000) 83-102.
[5] S.M. Fallat, C.R. Johnson, J.R. Torregrosa and A.M. Urbano, P-matrix
completions under weak symmetry assumptions, Linear Algebra Appl.,
312 (2000) 73-91.
[6] J. Bowers, J. Evers, L. Hogben, S. Shaner, K. Snider, and A. Wangsness,
On completion problems for various classes of P-matrices, Linear
Algebra Appl., 413 (2006) 342-354.
[7] J.Y. Choi, L.M. DeAlba, L. Hogben, B. Kivunge, S. Nordstrom, and M.
Shedenhelm, The nonnegative P0 -matrix completion problem, Electronic
Journal of Linear Algebra, 10 (2003) 46-59.
[8] J.Y. Choi, L.M. DeAlba, L. Hogben, M. Maxwell and A. Wangsness, The
P0-matrix completion problem, Electronic Journal of Linear Algebra, 9
(2002) 1-20.
[9] L. Hogben, Completions of M-matrix patterns, Linear Algebra Appl.,
285 (1998) 143-152.
[10] L. Hogben, Inverse M-matrix completions of patterns omitting some
diagonal positions, Linear Algebra Appl., 313 (2000) 173-192.
[11] L. Hogben, The symmetric M-matrix and symmetric inverse M-matrix
completion problems, Linear Algebra Appl., 353 (2002) 159-167.
[12] C. Mendes Ara'ujo, J.R. Torregrosa and A.M. Urbano, N-matrix completion
problem, Linear Algebra Appl., 372 (2003) 111-125.
[13] C. Mendes Ara'ujo, J.R. Torregrosa and A.M. Urbano, The N-matrix
completion problem under digraphs assumptions, Linear Algebra Appl.,
380 (2004) 213-225.
[14] C. Mendes Ara'ujo, J.R. Torregrosa, A.M. Urbano, The symmetric Nmatrix
completion problem, Linear Algebra Appl., 406 (2005) 235-252.
[15] C. R. Johnson, M. Lundquist, T. J. Lundy, J. S. Maybee, Deterministic
inverse zero-patterns, Diserete mathematics 113(2001) 211-236.
[16] L. Hogben, Graph theoretic methods for matrix completion problems,
Linear Algebra Appl., 328 (2001) 161-202.
[17] Gray Chartrand, Ping Zhang, Introduction to graph theory, Published by
the McGraw-Hill Companies, Inc, 2005.
[1] Gu-Fang Mou, Ting-Zhu Huang, The N10 -matrix completion problem, to
appear.
[2] T. Parthasathy, G. Ravindran, N-matrices, Linear Algebra Appl., 139
(1990) 89-102.
[3] Sheng-Wei Zhou, Ting-Zhu Huang, On Perron complements of inverse
N0-matrices, Linear Algebra Appl., 434 (2011) 2081-2088.
[4] L. DeAlba and L. Hogben, Completion problems of P-matrix patterns,
Linear Algebra Appl., 319 (2000) 83-102.
[5] S.M. Fallat, C.R. Johnson, J.R. Torregrosa and A.M. Urbano, P-matrix
completions under weak symmetry assumptions, Linear Algebra Appl.,
312 (2000) 73-91.
[6] J. Bowers, J. Evers, L. Hogben, S. Shaner, K. Snider, and A. Wangsness,
On completion problems for various classes of P-matrices, Linear
Algebra Appl., 413 (2006) 342-354.
[7] J.Y. Choi, L.M. DeAlba, L. Hogben, B. Kivunge, S. Nordstrom, and M.
Shedenhelm, The nonnegative P0 -matrix completion problem, Electronic
Journal of Linear Algebra, 10 (2003) 46-59.
[8] J.Y. Choi, L.M. DeAlba, L. Hogben, M. Maxwell and A. Wangsness, The
P0-matrix completion problem, Electronic Journal of Linear Algebra, 9
(2002) 1-20.
[9] L. Hogben, Completions of M-matrix patterns, Linear Algebra Appl.,
285 (1998) 143-152.
[10] L. Hogben, Inverse M-matrix completions of patterns omitting some
diagonal positions, Linear Algebra Appl., 313 (2000) 173-192.
[11] L. Hogben, The symmetric M-matrix and symmetric inverse M-matrix
completion problems, Linear Algebra Appl., 353 (2002) 159-167.
[12] C. Mendes Ara'ujo, J.R. Torregrosa and A.M. Urbano, N-matrix completion
problem, Linear Algebra Appl., 372 (2003) 111-125.
[13] C. Mendes Ara'ujo, J.R. Torregrosa and A.M. Urbano, The N-matrix
completion problem under digraphs assumptions, Linear Algebra Appl.,
380 (2004) 213-225.
[14] C. Mendes Ara'ujo, J.R. Torregrosa, A.M. Urbano, The symmetric Nmatrix
completion problem, Linear Algebra Appl., 406 (2005) 235-252.
[15] C. R. Johnson, M. Lundquist, T. J. Lundy, J. S. Maybee, Deterministic
inverse zero-patterns, Diserete mathematics 113(2001) 211-236.
[16] L. Hogben, Graph theoretic methods for matrix completion problems,
Linear Algebra Appl., 328 (2001) 161-202.
[17] Gray Chartrand, Ping Zhang, Introduction to graph theory, Published by
the McGraw-Hill Companies, Inc, 2005.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:60560", author = "Gu-Fang Mou and Ting-Zhu Huang", title = "The Partial Non-combinatorially Symmetric N10 -Matrix Completion Problem", abstract = "An n×n matrix is called an N1 0 -matrix if all principal minors are non-positive and each entry is non-positive. In this paper, we study the partial non-combinatorially symmetric N1 0 -matrix completion problems if the graph of its specified entries is a transitive tournament or a double cycle. In general, these digraphs do not have N1 0 -completion. Therefore, we have given sufficient conditions that guarantee the existence of the N1 0 -completion for these digraphs.
", keywords = "Matrix completion, matrix completion, N10 -matrix, non-combinatorially symmetric, cycle, digraph.", volume = "5", number = "2", pages = "164-5", }
