Abstract
This paper builds upon the results in the article “G-matrices, J-orthogonal matrices, and their sign patterns", Czechoslovak Math. J. 66 (2016), 653-670, by Hall and Rozloznik. A number of further general results on the sign patterns of the J-orthogonal matrices are proved. Properties of block diagonal matrices and their sign patterns are examined. It is shown that all 4 × 4 full sign patterns allow J-orthogonality. Important tools in this analysis are Theorem 2.2 on the exchange operator and Theorem 3.2 on the characterization of J-orthogonal matrices in the paper “J-orthogonal matrices: properties and generation", SIAM Review 45 (3) (2003), 504-519, by Higham. As a result, it follows that for n ≤4 all n×n full sign patterns allow a J-orthogonal matrix as well as a G-matrix. In addition, the 3 × 3 sign patterns of the J-orthogonal matrices which have zero entries are characterized.
References
[1] L. B. Beasley and D. J. Scully, Linear operators which preserve combinatorial orthogonality, Linear Algebra Appl. 201 (1994), 171-180.10.1016/0024-3795(94)90114-7Search in Google Scholar
[2] R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, Cambridge, 1991.10.1017/CBO9781107325708Search in Google Scholar
[3] R. A. Brualdi and B. L. Shader, Matrices of Sign-Solvable Linear Systems, Cambridge University Press, Cambridge, 1995.10.1017/CBO9780511574733Search in Google Scholar
[4] C. A. Eschenbach, F. J. Hall, D. L. Harrell, and Z. Li, When does the inverse have the same sign pattern as the transpose?, Czechoslovak Math. J. 49 (1999), 255-275.10.1023/A:1022496101277Search in Google Scholar
[5] M. Fiedler, Notes on Hilbert and Cauchy matrices, Linear Algebra Appl. 432 (2010), 351-356.10.1016/j.laa.2009.08.014Search in Google Scholar
[6] M. Fiedler and F. J. Hall, G-matrices, Linear Algebra Appl. 436 (2012), 731-742.10.1016/j.laa.2011.08.001Search in Google Scholar
[7] E. J. Grimme, D. C. Sorensen, and P. Van Dooren, Model reduction of state space systems via an implicitly restarted Lanczos method, Numerical Algorithms, 12(1-2) (1996), 1-31.10.1007/BF02141739Search in Google Scholar
[8] F. J. Hall and Z. Li, Sign pattern matrices, Chapt. 42 in L. Hogben (Ed.), Handbook of Linear Algebra, 2nd ed., Chapman and Hall/CRC Press, Boca Raton, 2013.10.1201/b16113-48Search in Google Scholar
[9] F. J. Hall and M. Rozloznik, G-matrices, J-orthogonal matrices, and their sign patterns, Czechoslovak Math. J. 66 (2016), 653-670.10.1007/s10587-016-0284-8Search in Google Scholar
[10] N. J. Higham, J-orthogonal matrices: properties and generation, SIAM Review 45 (3) (2003), 504-519.10.1137/S0036144502414930Search in Google Scholar
[11] R. Horn and C.R. Johnson, Matrix analysis, 2nd ed., Cambridge University Press, Cambridge, 2013.Search in Google Scholar
[12] M. Stewart and G.W. Stewart, On hyperbolic triangularization: stability and pivoting, SIAM J. Matrix Anal. Appl. 19 (1998), no. 4, 847-860.Search in Google Scholar
[13] M. Stewart and P. Van Dooren, On the factorization of hyperbolic and unitary transformations into rotations, SIAM J. Matrix Anal. Appl. 27 (2006), no. 3, 876-890.Search in Google Scholar
[14] C. Waters, Sign patterns that allow orthogonality, Linear Algebra Appl. 235 (1996), 1-13.10.1016/0024-3795(94)00098-0Search in Google Scholar
© 2017
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.