Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.     

Linear transformations on matrices: the invariance of sets of ranks

Let R_E denote the set of all m × n matrices over an algebraically closed field F whose ranks lie in the set E, where E is a subset of {1,2,…,m}. Let T be a linear transformation which maps R_E into itself. Under some restrictions on E, or when T is nonsingular, there are nonsingular matrices U and V such that T(A) = UAV for every m × n matrix A.     

Large spaces of symmetric matrices of bounded rank are decomposable ∗

Let k and n be positive integers such that kn. Let S_n (F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n (F) is said to be a k-subspace if rank Ak for every A?L.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n (F) is decomposable if there exists in Fⁿ a subspace W of dimension n?r such that x^tAx=0 for every x?W A?L.

We show here, under some mild assumptions on k n and F, that every k∥-subspace of S_n (F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n .     

Possible numbers of positive entries of imprimitive nonnegative matrices

Zhan, X., Extremal numbers of positive entries of imprimitive nonnegative matrix, Linear Algebra Appl. (in press) has determined the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices with a given imprimitivity index. Let σ( A ) denote the number of positive entries of a matrix A. Let M(n,?k) and m(n,?k) denote the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices of order n with a given imprimitivity index k, respectively. In this article, we prove that for any positive integer d with m(n,k)≤ d?≤?M(n,k), there exists an n?×?n irreducible nonnegative matrix A with imprimitivity index k such that?σ?(A)=d.     

Idempotence preserving maps between matrix spaces

Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽₃. Let M _n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n idempotent matrices and GL _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n invertible matrices. Let Φ_𝔽(n,?m) denote the set of all maps from M _n (𝔽) to M _m (𝔽) satisfying A???λB?∈?P _n (𝔽)???φ(A)???λφ(B)?∈?P _m (𝔽) for every A,?B?∈?M _n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ_𝔽(n,?m), then there exists T?∈?GL _m (𝔽) such that φ(A)?=?T?[A???I _p ?⊕?A ^t ???I _q ?⊕?0]T?^?1 for every A?∈?M _n (𝔽), where I ₀?=?0. This improves the results of some related references.     

A note on k-potence preserving maps

Let M_n be the algebra of all n×n complex matrices and Γ_n the set of all k-potent matrices in M_n. Suppose ?:M_n→M_n is a map satisfying A-λB∈Γ_n implies ?(A)-λ?(B)∈Γ_n, where A, B∈M_n, λ∈C. Then either ? is of the form ?(A)=cTAT^-1, A∈M_n, or ? is of the form ?(A)=cTA^tT^-1, A∈M_n, where T∈M_n is an invertible matrix, c∈C satisfies c^k=c.     

Ranked solutions of AXC=B and AX=B

Let GF(pⁿ) denote the finite field of pⁿ elements, p odd. Let A be an s×m matrix of rank ?, B be an s×t matrix of rank β, and C be an f×t matrix of rank v. This paper discusses the number of m×f matrices X of rank k over GF(pⁿ) which are solutions to the matric equations AXC=B or AX=B.     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

A linear-algebra problem from algebraic coding theory

Given an m×n matrix M over E=GF(q^t) and an ordered basis A={z₁,…,z_t} for field E over K=GF(q), expand each entry of M into a t×1 vector of coordinates of this entry relative to A to obtain an mt×n matrix M¹ with entries from the field K. Let r=rank(M) and r¹=rank(M¹). We show that r?r¹?min{rt,n}, and we determine the number b(m,n,r,r¹,q,t) of m×n matrices M of rank r over GF(q^t) with associated mt×n matrix M¹ of rank r¹ over GF (q).     

The primitive Boolean matrices with the second largest scrambling index by Boolean rank

The scrambling index of an n × n primitive Boolean matrix A is the smallest positive integer k such that A ^k (A ^T) ^k = J, where A ^T denotes the transpose of A and J denotes the n×n all ones matrix. For an m×n Boolean matrix M, its Boolean rank b(M) is the smallest positive integer b such that M = AB for some m × b Boolean matrix A and b×n Boolean matrix B. In 2009, M. Akelbek, S. Fital, and J. Shen gave an upper bound on the scrambling index of an n×n primitive matrix M in terms of its Boolean rank b(M), and they also characterized all primitive matrices that achieve the upper bound. In this paper, we characterize primitive Boolean matrices that achieve the second largest scrambling index in terms of their Boolean rank.     

Rees algebras with minimal multiplicity

Let (R,m) be a local ring. Let S_M denote the Rees algebra S=R[m^rt] localized at its unique maximal homogeneous ideal M=(m,m^rt). Let TN denote the extended Rees algebra T= R[m^rt, t^-1] localized at its unique maximal homogeneous idea N= (t^?1,m,m^r). Multiplicity formulas are developedfor S_M and T_N. These are used to find necessaIy and sufficient conditions on a Cohen-Macaulay local ring (R,m) and r so that S_M and T_N are Cohen-Macaulay with minimal multiplicity     

Products of reflections in the general linear group over a division ring

Let F be a division ring and A?GL_n(F). We determine the smallest integer k such that A admits a factorization A=R₁R₂?R_k?1B, where R₁,…,R_k?1 are reflections and B is such that rank(B?I_n)=1. We find that, apart from two very special exceptional cases, k=rank(A?I_n). In the exceptional cases k is one larger than this rank. The first exceptional case is the matrices A of the form I_m⊕αI_n?m where n?m?2, α≠?1, and α belongs to the center of F. The second exceptional case is the matrices A satisfying (A?I_n)²=0, rank(A?I_n)?2 in the case when char F≠2 only. This result is used to determine, in the case when F is commutative, the length of a matrix A?GL_n(F) with detA=±1 with respect to the set of all reflections in GL_n(F).     

Lower bound inequalities for norms of symmetrized tensor powers

Let V be a complex inner product space of positive dimension m with inner product 〈·,·〉, and let T_n(V) denote the set of all n-linear complex-valued functions defined on V×V×?×V (n-copies). By S_n(V) we mean the set of all symmetric members of T_n(V). We extend the inner product, 〈·,·〉, on V to T_n(V) in the usual way, and we define multiple tensor products A₁⊗A₂⊗?⊗A_n and symmetric products A₁·A₂?A_n, where q₁,q₂,…,q_n are positive integers and A_i∈T_qi(V) for each i, as expected. If A∈S_n(V), then A^k denotes the symmetric product A·A?A where there are k copies of A. We are concerned with producing the best lower bounds for ‖A^k‖², particularly when n=2. In this case we are able to show that ‖A^k‖² is a symmetric polynomial in the eigenvalues of a positive semi-definite Hermitian matrix, M_A, that is closely related to A. From this we are able to obtain many lower bounds for ‖A^k‖². In particular, we are able to show that if ω denotes 1/r where r is the rank of M_A, and , then

     

Surjections on Grassmannians preserving pairs of elements with bounded distance

Let m and k be two fixed positive integers such that m>k?2. Let V be a left vector space over a division ring with dimension at least m+k+1. Let G_m(V) be the Grassmannian consisting of all m-dimensional subspaces of V. We characterize surjective mappings T from G_m(V) onto itself such that for any A,B in G_m(V), the distance between A and B is not greater than k if and only if the distance between T(A) and T(B) is not greater than k.     

Constructing Maximal Commutative Subalgebras of Matrix Rings in Small Dimensions

Let k denote an algebraically closed field of arbitrary characteristic. Let C denote the set of all commutative, finite dimensional, local k-algebras of the form (B, m, k) with i(m) ?2. Here i(m) denotes the index of nilpotency of the maximal ideal m. A Akalgebra (R, J,k)∈L is called a (c₁-construction if there exists (B, m, k)∈ £ ? {(k, (0), k)} and a finitely generated, faithful B-module N such that R,?B?(the idealization of N). (R.J.k) is called a (c_2::-construction if there exist a (B,m k)∈ L, a positive integer p $ge;2 and a nonzero z £ S_B(the socle of B) such that R?B[x]/(mX, X^p- z). Let M_n×n(K) denote the set of all n x n matrices, over k with n≥2. Let .M_n(k) denote the set of all maximal, commutative A;-subalgebras of M_n×n(k). In this paper, we show any (R J, k) ∈£?M_n;(k) with n>5 is a C₁ or C₂ -construction except for one isomorphism class. The one exception occurs when n = 5.     

Linear maps preserving M–P inverses of matrices between matrix spaces over fields

Suppose F is a field of characteristic not 2. Let n and m be two arbitrary positive integers with n≥2. We denote by M _n (F) and S _n (F) the space of n×n full matrices and the space of n×n symmetric matrices over F, respectively. All linear maps from S _n (F) to M _m (F) preserving M–P inverses of matrices are characterized first, and thereby all linear maps from S _n (F) (M _n (F)) to S _m (F) (M _m (F)) preserving M–P inverses of matrices are characterized, respectively.     

Domino tilings of rectangles with fixed width

Let t(k,n) denote the number of ways to tile a 1 × n rectangle with 1 × 2 rectangles (called dominoes). We show that for each fixed k the sequence t_k=(t(k,0), t(k,1),…) satisfies a difference equation (linear, homogeneous, and with constant coefficients). Furthermore, a computational method is given for finding this difference equation together with the initial terms of the sequence. This gives rise to a new way to compute t(k,n) which differs completely with the known Pfaffian method. The generating function of t_k is a rational function F_k, and F_k is given explicitly for k=1,…,8. We end with some conjectures concerning the form of F_k based on our computations.     

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.