Linear operators that preserve Boolean rank of Boolean matrices

The Boolean rank of a nonzero m × n Boolean matrix A is the minimum number k such that there exist an m× k Boolean matrix B and a k × n Boolean matrix C such that A = BC. In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and 2. In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and k for some 1 < k ? m.     

Linear maps that preserve commuting pairs of matrices

Let L be a linear map on the space of n by n matrices with entries in an algebraically closed field of characteristic 0. In this article we characterize all non-singular L with the property that AB = BA implies L(A)L(B) = L(B)L(A).     

Linear maps that preserve singular and nonsingular matrices

Let M (n,K) be the algebra of n × n matrices over an algebraically closed field K and T:M (n,K)→M (n,K) a linear transformation with the property that T maps nonsingular (singular) matrices to nonsingular (singular) matrices. Using some elementary facts from commutative algebra we show that T is nonsingular and maps singular matrices to singular matrices (T is nonsingular or T maps all matrices to singular matrices). Using these results we obtain Marcus and Moyl's characterization [T(x) = UXVorU^tXV for fixed U and V] from a result of Dieudonné's. Examples are given to show the hypothesis of algebraic closure in necessary.     

Linear maps that preserve matrices annihilated by a polynomial

Let M_n be the algebra of n×n matrices over an algebraically closed field of characteristic zero. Let f(x) be a polynomial over $\text{F}$ with at least two distinct roots. Then all nonsingular linear maps L:M_n→M_n that map matrix roots of F(x)=0 into matrix roots of f(x}=0 are found.     

Continuous maps on triangular matrices that preserve commutativity

We investigate continuous injective maps defined on the space of upper triangular matrices over a field F that preserve commutativity of matrices in both directions. We show that every such map is a composition of either an inner automorphism and a locally polynomial map or the two latter and one more automorphism of upper triangular matrices.     

Linear transformations on symmetric matrices that preserve commutativity

It is shown that if a nonsingular linear transformation T on the space of n-square real symmetric matrices preserves the commutativity, where n ?3, then T(A) = λQAQ^t + Q(A)I_n for all symmetric matricesA, for some scalar λ, orthogonal matrix Q, and linear functional Q.     

Linear transformations on symmetric matrices that preserve the permanent

It is shown that if a linear transformation T on the space of n-square symmetric matrices over any subfield of the real field preserves the permanent, where n ? 3, then T(A)= ± PAP^t for all symmetric matrices A and a fixed generalized permutation matrix P with per P= ± 1.     

Linear transformations that preserve certain positivity classes of matrices

For each of several S ? R^n,n, those linear transformations $\text{L}\text{: R}{}^{\mathrm{n,n}}\text{→ R}{}^{\mathrm{n,n}}$ which map S onto S are characterized. Each class is a familiar one which generalizes the notion of positivity to matrices. The classes include: the matrices with nonnegative principal minors, the M-matrices, the totally nonnegative matrices, the D-stable matrices, the matrices with positive diagonal Lyapunov solutions, and the H-matrices, as well as other related classes. The set of transformations is somewhat different from case to case, but the strategy of proof, while differing in detail, is similar.     

Linear operators strongly preserving r-cyclic matrices over semirings

We say that A is an r-cyclic matrix if A^r=I. We investigate the structure of linear operators on matrices over antinegative semirings that map r-cyclic matrices to r-cyclic matrices and non r-cyclic matrices to non r-cyclic matrices.     

Linear operators that preserve maximal column rank of boolean matrices

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.     

Linear operators that preserve maximal column ranks of nonnegative integer matrices

The maximal column rank of an by matrix over a semiring is the maximal number of the columns of which are linearly independent. We characterize the linear operators which preserve the maximal column ranks of nonnegative integer matrices.

     

Linear maps on operator algebras that preserve elements annihilated by a polynomial

In this paper some purely algebraic results are given concerning linear maps on algebras which preserve elements annihilated by a polynomial of degree greater than 1 and with no repeated roots and applied to linear maps on operator algebras such as standard operator algebras, von Neumann algebras and Banach algebras. Several results are obtained that characterize such linear maps in terms of homomorphisms, anti-homomorphisms, or, at least, Jordan homomorphisms.

     

Linear operators that preserve pairs of matrices which satisfy extreme rank properties

A pair of m×n matrices (A,B) is called rank-sum-maximal if rank(A+B)=rank(A)+rank(B), and rank-sum-minimal if rank(A+B)=|rank(A)−rank(B)|. We characterize the linear operators that preserve the set of rank-sum-minimal matrix pairs, and the linear operators that preserve the set of rank-sum-maximal matrix pairs over any field with at least min(m,n)+2 elements and of characteristic not 2.     

Maps on matrices that preserve the spectrum

Let T be a continuous map of the space of complex n×n matrices into itself satisfying T(0)=0 such that the spectrum of T(x)-T(y) is always a subset of the spectrum of x-y. There exists then an invertible n×n matrix u such that either T(a)=uau^-1 for all a or T(a)=ua^tu^-1 for all a. We arrive at the same conclusion by supposing that the spectrum of x-y is always a subset of the spectrum of T(x)-Tt(y), without the continuity assumption on T.     

Linear equations and interpolation in Boolean algebra

The authors generalize the classical interpolation formula for Boolean functions of n variables. A characterization of all interpolating systems with 2ⁿ elements is obtained. The methods of proof used are intimately related to the solution of linear Boolean equations.     

Digital maps,matrices and computer algebra

The way in which computer algebra systems, such as Maple®, have made the study of complex problems accessible to undergraduate mathematicians with modest computational skills is illustrated by some large matrix calculations, which arise from representing the Earth's surface by digital elevation models. Such problems are often considered to lie in the field of computer mapping and thus addressed by geographical information systems. The problems include simple identification of local maximum points, visualization by cross-sectional profiles, contour maps and three-dimensional views, consideration of the visual impact of the placement of large buildings and issues arising from reservoir creation. Motion through a virtual landscape can be simulated by an animation facility. This approach has been successful with first year students: the ‘real world’ problems considered are more accessible than many alternatives, and the attraction of using large matrices is retained.     

On generalized circulants over a Boolean algebra

For any given positive integer n, we give a necessary and sufficient condition for an n × nr-circulant (generalized circulant) over the Boolean algebra B = {0, 1} to be idempotent, and we present an algorith to obtain all n × n indempotent r-circulants over B for r = 0, 1,…,n ? 1.