If S is a semiring of nonnegative reals, which linear operators T on the space of $m \times n$ ma... more If S is a semiring of nonnegative reals, which linear operators T on the space of $m \times n$ matrices over S preserve the column rank of each matrix\ulcorner Evidently if P and Q are invertible matrices whose inverses have entries in S, then $T : X \longrightarrow PXQ$ is a column rank preserving, linear operator. Beasley and Song obtained some characterizations of column rank preserving linear operators on the space of $m \times n$ matrices over $Z_+$, the semiring of nonnegative integers in [1] and over the binary Boolean algebra in [7] and [8]. In [4], Beasley, Gregory and Pullman obtained characterizations of semiring rank-1 matrices and semiring rank preserving operators over certain semirings of the nonnegative reals. We considers over certain semirings of the nonnegative reals. We consider some results in [4] in view of a certain column rank instead of semiring rank.
ABSTRACT An n× nmatrix Ais called convertible if there is an n× n(1, -1)-matrix Hsuch that per A=... more ABSTRACT An n× nmatrix Ais called convertible if there is an n× n(1, -1)-matrix Hsuch that per A= det(H∘A) where H ∘ Adenotes the Hadamard product of Hand A. A convertible (0,l)-matrix is called extremal if replacing any zero entry with a 1 breaks the convertibility. In this paper some properties of nonnegative convertible matrices are investigated and some classes of extremal convertible (0,1)-matrices are obtained.
A square real matrix Ais called convertible if there is a matrix  obtained from Aby affixing ± s... more A square real matrix Ais called convertible if there is a matrix  obtained from Aby affixing ± sings to entries of Aso that per A=detÂ. A convertible (0,1)-matrix with total support is called maximal convertible if it is fully indecomposable and no matrix obtained from Aby replacing a0 with a1 is convertible. In this paper, the existence of maximal convetible matrices with exactly r1's for each integer rwith 4n−4≤r≤(n2+3n−2)/2 is proved.
We determine the minimum permanents and minimizing matrices on the face of ω3+n, for the fully in... more We determine the minimum permanents and minimizing matrices on the face of ω3+n, for the fully indecomposable (0,1) matrices of order 3+n, which include an identity submatrix of order n .
ABSTRACT This paper concerns two notions of column rank of matrices over semirings; column rank a... more ABSTRACT This paper concerns two notions of column rank of matrices over semirings; column rank and maximal column rank. These two notions are the same over fields but differ for matrices over certain semirings. We determine how much the maximal column rank is different from the column ran for all m×n matrices over many semirings. We also characterize the linear operators which preserve the maximal column rank of Boolean matrices.
If S is a semiring of nonnegative reals, which linear operators T on the space of $m \times n$ ma... more If S is a semiring of nonnegative reals, which linear operators T on the space of $m \times n$ matrices over S preserve the column rank of each matrix\ulcorner Evidently if P and Q are invertible matrices whose inverses have entries in S, then $T : X \longrightarrow PXQ$ is a column rank preserving, linear operator. Beasley and Song obtained some characterizations of column rank preserving linear operators on the space of $m \times n$ matrices over $Z_+$, the semiring of nonnegative integers in [1] and over the binary Boolean algebra in [7] and [8]. In [4], Beasley, Gregory and Pullman obtained characterizations of semiring rank-1 matrices and semiring rank preserving operators over certain semirings of the nonnegative reals. We considers over certain semirings of the nonnegative reals. We consider some results in [4] in view of a certain column rank instead of semiring rank.
ABSTRACT An n× nmatrix Ais called convertible if there is an n× n(1, -1)-matrix Hsuch that per A=... more ABSTRACT An n× nmatrix Ais called convertible if there is an n× n(1, -1)-matrix Hsuch that per A= det(H∘A) where H ∘ Adenotes the Hadamard product of Hand A. A convertible (0,l)-matrix is called extremal if replacing any zero entry with a 1 breaks the convertibility. In this paper some properties of nonnegative convertible matrices are investigated and some classes of extremal convertible (0,1)-matrices are obtained.
A square real matrix Ais called convertible if there is a matrix  obtained from Aby affixing ± s... more A square real matrix Ais called convertible if there is a matrix  obtained from Aby affixing ± sings to entries of Aso that per A=detÂ. A convertible (0,1)-matrix with total support is called maximal convertible if it is fully indecomposable and no matrix obtained from Aby replacing a0 with a1 is convertible. In this paper, the existence of maximal convetible matrices with exactly r1's for each integer rwith 4n−4≤r≤(n2+3n−2)/2 is proved.
We determine the minimum permanents and minimizing matrices on the face of ω3+n, for the fully in... more We determine the minimum permanents and minimizing matrices on the face of ω3+n, for the fully indecomposable (0,1) matrices of order 3+n, which include an identity submatrix of order n .
ABSTRACT This paper concerns two notions of column rank of matrices over semirings; column rank a... more ABSTRACT This paper concerns two notions of column rank of matrices over semirings; column rank and maximal column rank. These two notions are the same over fields but differ for matrices over certain semirings. We determine how much the maximal column rank is different from the column ran for all m×n matrices over many semirings. We also characterize the linear operators which preserve the maximal column rank of Boolean matrices.
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