Sign Pattern Matrices

We look at Bohemians, specifically those with population {−1, 0, +1} and sometimes {0,1,i, − 1, − i}. More, we specialize the matrices to be upper Hessenberg Bohemian. From there, focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give useful bounds on their height, and conjecture an accurate asymptotic formula. The lower bound for the maximal characteristic height is exponential in the order of the matrix; in contrast, the height of the matrices remains constant. We give theorems about the numbers of normal matrices and the numbers of stable matrices in these families.

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“…Matrices with population P = {−1,0,1} occur naturally as exemplars of "signpattern matrices": see [6] and [21]. For early theorems, see [25].…”

Section: Introductionmentioning

confidence: 99%

Upper Hessenberg and Toeplitz Bohemians

Chan

Corless

González-Vega³

et al. 2020

Linear Algebra and its Applications

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“…It is well-known that an n × n sign pattern matrix A is sign singular if and only if A has no "composite cycle" of length n. The reader is referred to [3] or [8] for more information on sign pattern matrices.…”

Section: Theorem 13 (I) For a Xed Signature Matrix J The Set Of Almentioning

confidence: 99%

Sign patterns of J-orthogonal matrices

Hall

Parnass

et al. 2017

Special Matrices

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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 × 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 ≤ all n×n full sign patterns allow a J-orthogonal matrix as well as a G-matrix. In addition, the × sign patterns of the J-orthogonal matrices which have zero entries are characterized.

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“…The sign pattern of matrix has important applications in the qualitative economics (see [4,16,17,20,21,23]). The monograph of Brualdi and Shader introduces many results on S 2 NS-matrices (see [4]).…”

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confidence: 99%

Representations and sign pattern of the group inverse for some block matrices

Sun

Wang

et al. 2015

ELA

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Abstract. Let M =A B C 0 be a complex square matrix where A is square. When BCB Ω = 0, rank(BC) = rank(B) and the group inverse of B Ω AB Ω 0 CB Ω 0 exists, the group inverse of M exists if and only if rank(BC +. In this case, a representation of M # in terms of the group inverse and Moore-Penrose inverse of its subblocks is given. Let A be a real matrix. The sign pattern of A is a (0, +, −)-matrix obtained from A by replacing each entry by its sign. The qualitative class of A is the set of the matrices with the same sign pattern as A, denoted by Q(A). The matrix A is called S 2 GI, if the group inverse of each matrix A ∈ Q(A) exists and its sign pattern is independent of A. By using the group inverse representation, a necessary and sufficient condition for a real block matrixwhere A is square, ∆ 1 and ∆ 2 are invertible, Y 1 and Y 2 are sign orthogonal.

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Sign Pattern Matrices

Cited by 6 publications

References 163 publications

Upper Hessenberg and Toeplitz Bohemians

Upper Hessenberg and Toeplitz Bohemians

Sign patterns of J-orthogonal matrices

Representations and sign pattern of the group inverse for some block matrices

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