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2. Sign patterns of J-orthogonal matrices

Author

Hall Frank J., Li Zhongshan, Parnass Caroline T., and Rozložník Miroslav

Subjects
g-matrix, j-orthogonal matrix, sign pattern matrix, sign patterns that allow j-orthogonality, Mathematics, QA1-939
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.

Published
2017
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4. Essential sign change numbers of full sign pattern matrices

Author

Chen Xiaofeng, Fang Wei, Gao Wei, Gao Yubin, Jing Guangming, Li Zhongshan, Shao Yanling, and Zhang Lihua

Subjects
sign pattern (matrix), full sign pattern, minimum rank, sign change number, essential row sign change number, essential column sign change number, sign vectors of a list of polynomials, Mathematics, QA1-939
Abstract

A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0} and a sign vector is a vector whose entries are from the set {+, −, 0}. A sign pattern or sign vector is full if it does not contain any zero entries. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. The notions of essential row sign change number and essential column sign change number are introduced for full sign patterns and condensed sign patterns. By inspecting the sign vectors realized by a list of real polynomials in one variable, a lower bound on the essential row and column sign change numbers is obtained. Using point-line confiurations on the plane, it is shown that even for full sign patterns with minimum rank 3, the essential row and column sign change numbers can differ greatly and can be much bigger than the minimum rank. Some open problems concerning square full sign patterns with large minimum ranks are discussed.

Published
2016
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