Linear transformations which preserve fixed rank

Let M_{m, n}(F) denote the set of all m×n matrices over the algebraically closed field F. Let T denote a linear transformation, T:M_{m, n}(F)→M_{m, n}(F). Theorem: If max(m, n)?2k?2, k?1, and T preserves rank k matrices [i.e.?(A)=k implies ?(T(A))=k], then there exist nonsingular m×m and n×n matrices U and V respectively such that either (i) T:A→UAV for all A?M_{m, n}(F), or (ii) m=n and T:A→UA^tV for all A?M_n(F), where A^t denotes the transpose of A.     

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.     

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.     

Higher-order recurrences for Bernoulli numbers

Euler's well-known nonlinear relation for Bernoulli numbers, which can be written in symbolic notation as n(B₀+B₀)=−nBn−1−(n−1)Bn, is extended to n(Bk₁+?+Bkm) for m?2 and arbitrary fixed integers k₁,…,km?0. In the general case we prove an existence theorem for Euler-type formulas, and for m=3 we obtain explicit expressions. This extends the authors' previous work for m=2.     

On simple factorization of invertible matrices

Let I be an ideal of a ring R. We say that R is a generalized I-stable ring provided that aR+bR=R with a?∈?1+I,b?∈?R implies that there exists a y?∈?R such that a+by?∈?K(R), where K(R)={x?∈?R?∣?? s, t?∈?R such that sxt=1}. Let R be a generalized I-stable ring. Then every A?∈?GL_n (I) is the product of 13n?12 simple matrices. Furthermore, we prove that A is the product of n simple matrices if I has stable rank one. This generalizes the results of Vaserstein and Wheland on rings having stable rank one.     

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.     

On classical adjoint-commuting mappings between matrix algebras

Let F be a field and let m and n be integers with m,n?3. Let M_n denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:M_n→M_m that satisfy one of the following conditions:

1.

|F|=2 or |F|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for all A,B∈M_n and α∈F with ψ(I_n)≠0.

2.

ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈M_n.

Here, adjA denotes the classical adjoint of the matrix A, and I_n is the identity matrix of order n. We give examples showing the indispensability of the assumption ψ(I_n)≠0 in our results.     

A common property of R(E, F) and B(ℝ n , ℝ m ) and a new method for seeking a path to connect two operators

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, and R(E, F) the set of all operators in B(E, F) with finite rank. It is well-known that B(? ⁿ ) is a Banach space as well as an algebra, while B(? ⁿ , ? ^m ) for m ≠ n, is a Banach space but not an algebra; meanwhile, it is clear that R(E, F) is neither a Banach space nor an algebra. However, in this paper, it is proved that all of them have a common property in geometry and topology, i.e., they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces). Let Σ _r be the set of all operators of finite rank r in B(E, F) (or B(? ⁿ , ? ^m )). In fact, we have that 1) suppose Σ _r ∈ B(? ⁿ , ? ^m ), and then Σ _r is a smooth and path-connected submanifold of B(? ⁿ , ? ^m ) and dimΣ _r = (n + m)r ? r ², for each r ∈ [0, min{n,m}; if m ≠ n, the same conclusion for Σ _r and its dimension is valid for each r ∈ [0, min{n, m}]; 2) suppose Σ _r ∈ B(E, F), and dimF = ∞, and then Σ _r is a smooth and path-connected submanifold of B(E, F) with the tangent space T _A Σ _r = {B ∈ B(E, F): BN(A) ? R(A)} at each A ∈ Σ _r for 0 ? r ? ∞. The routine methods for seeking a path to connect two operators can hardly apply here. A new method and some fundamental theorems are introduced in this paper, which is development of elementary transformation of matrices in B(? ⁿ ), and more adapted and simple than the elementary transformation method. In addition to tensor analysis and application of Thom’s famous result for transversility, these will benefit the study of infinite geometry.     

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 relationship between stochastic matrices and eigenvalues of products

Let {B₁,…,B_n} be a set of n rank one n×n row stochastic matrices. The next two statements are equivalent: (1) If A is an n×n nonnegative matrix, then 1 is an eigenvalue ofB_kA for each k=1,…,n if and only if A is row stochastic. (2) The n×n row stochastic matrix S whose kth row is a row of B_k for k=1,…,n is nonsingular. For any set {B₁, B₂,…, B_p} of fewer than n row stochastic matrices of order n×n and of any rank, there exists a nonnegative n×n matrix A which is not row stochastic such that 1 is eigenvalue of every B_kA, k=1,…,p.     

A result about determinantal divisors

Let Rbe a principal ideal ringR_n the ring of n× nmatrices over R, and d_k (A) the kth determinantal divisor of Afor 1 ? k? n, where Ais any element of R_n , It is shown that if A,BεR_n , det(A) det(B:) ≠ 0, then d_k (AB) ≡ 0 mod d_k (A) d_k (B). If in addition (det(A), det(B)) = 1, then it is also shown that d_k (AB) = d_k (A) d_k (B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants.     

Simultaneous triangularization of a pair of matrices whose commutator has rank two

Let A,B be n×n matrices with entries in an algebraically closed field F of characteristic zero, and let C=AB?BA. It is shown that if C has rank two and AⁱB^jC^k is nilpotent for 0?i, j?n?1, 1?k?2, then A, B are simultaneously triangularizable over F. An example is given to show that this result is in some sense best possible.     

On polyvectors of vector spaces and hyperplanes of projective Grassmannians

We investigate relationships between polyvectors of a vector space V, alternating multilinear forms on V, hyperplanes of projective Grassmannians and regular spreads of projective spaces. Suppose V is an n-dimensional vector space over a field F and that A_n-1,k(F) is the Grassmannian of the (k − 1)-dimensional subspaces of PG(V) (1  ? k ? n − 1). With each hyperplane H of A_n-1,k(F), we associate an (n − k)-vector of V (i.e., a vector of ∧^n−kV) which we will call a representative vector of H. One of the problems which we consider is the isomorphism problem of hyperplanes of A_n-1,k(F), i.e., how isomorphism of hyperplanes can be recognized in terms of their representative vectors. Special attention is paid here to the case n = 2k and to those isomorphisms which arise from dualities of PG(V). We also prove that with each regular spread of the projective space PG(2k-1,F), there is associated some class of isomorphic hyperplanes of the Grassmannian A_2k-1,k(F), and we study some properties of these hyperplanes. The above investigations allow us to obtain a new proof for the classification, up to equivalence, of the trivectors of a 6-dimensional vector space over an arbitrary field F, and to obtain a classification, up to isomorphism, of all hyperplanes of A_5,3(F).     

Monotonicity of permanents of certain doubly stochastic circulant matrices

Let p_k(A), k=2,…,n, denote the sum of the permanents of all k×k submatrices of the n×n matrix A. A conjecture of Ðokovi?, which is stronger than the famed van der Waerden permanent conjecture, asserts that the functions p_k((1?θ)J_n+;θA), k=2,…, n, are strictly increasing in the interval 0?θ?1 for every doubly stochastic matrix A. Here J_n is the n×n matrix all whose entries are equal $\text{1}\text{n}$. In the present paper it is proved that the conjecture holds true for the circulant matrices A=αI_n+ βP_n, α, β?0, α+;β=1, and $\text{A=}\text{(nJ}{}_{\mathrm{n}}\text{?I}{}_{\mathrm{n}}\text{?P}{}_{\mathrm{n}}\text{)}\text{(n?2)}$, where I_n and P_n are respectively the n×n identify matrix and the n×n permutation matrix with 1's in positions (1,2), (2,3),…, (n?1, n), (n, 1).     

A new characterization of dual bases in finite fields and its applications

We propose a new characterization of dual bases in finite fields. Let A=(α₁,…,α_n) be a basis of F over F_q and its dual basis B=(β₁,…,β_n) with the transition matrix C∈GL_n(F_q) such that (β₁,…,β_n)=(α₁,…,α_n)C. We show that holds for all 1?k?n, where T_k∈M_n(F_q) satisfies α_k(α₁,…,α_n)=(α₁,…,α_n)T_k. Conversely, suppose F=F_q(α_k^′) and for some 1?k^′?n and G∈GL_n(F_q), then B is equivalent to (α₁,…,α_n)G. As applications, we can construct the dual basis of a given basis A or determine whether the dual basis of A satisfies the desired conditions from T_k. This generalizes the results obtained by Liao and Sun for normal bases. Furthermore, we give a simple proof of the theorem of Gollmann, Wang and Blake for polynomial bases.     

APPLICATIONS OF CLIFFORD ALGEBRAS TO INVOLUTIONS AND QUADRATIC FORMS

ABSTRACT

Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form ? ∈ I ²(F) such that dim ? = 8, [C(?)] = [D], and ? does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)/F is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution σ, then σ is hyperbolic if and only if σ _K(A) is hyperbolic. Finally, let σ be a decomposable orthogonal involution on the algebra M _{2 ^m} (K). In the case m ≤ 5 we give another proof of the fact that σ is a Pfister involution. If m ≥ 2 ^n?2 ? 2 and n ≥ 5, we show that q _σ ∈ I ⁿ (K), where q _σ is a quadratic form corresponding to σ. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms.     

Some intersection theorems

LetL(A) be the set of submatrices of anm×n matrixA. ThenL(A) is a ranked poset with respect to the inclusion, and the poset rank of a submatrix is the sum of the number of rows and columns minus 1, the rank of the empty matrix is zero. We attack the question: What is the maximum number of submatrices such that any two of them have intersection of rank at leastt? We have a solution fort=1,2 using the followoing theorem of independent interest. Letm(n,i,j,k) = max(|F|;|G|), where maximum is taken for all possible pairs of families of subsets of ann-element set such thatF isi-intersecting,G isj-intersecting andF ansd,G are cross-k-intersecting. Then fori≤j≤k, m(n,i,j,k) is attained ifF is a maximali-intersecting family containing subsets of size at leastn/2, andG is a maximal2k?i-intersecting family. Furthermore, we discuss and Erd?s-Ko-Rado-type question forL(A), as well.     

Semistability of Frobenius Direct Image of Representations of Cotangent Bundles

Let k be an algebraically closed field of characteristic p > 0, X a smooth projective variety over k with a fixed ample divisor H, F_X : X → X the absolute Frobenius morphism on X. Let E be a rational GL_n(k)-bundle on X, and ρ : GL_n(k) → GL_m(k) a rational GL_n(k)-representation of degree at most d such that ρ maps the radical RGL_n(k)) of GL_n(k) into the radical R(GL_m(k)) of GL_m(k). We show that if \(F_X^{N*}(E)\) is semistable for some integer \(N \ge {\max {_{0 < r < m}}}(_r^m) \cdot {\log _p}(dr)\), then the induced rational GL_m(k)-bundle E(GL_m(k)) is semistable. As an application, if dimX = n, we get a sufficient condition for the semistability of Frobenius direct image \(F_{X*}(\rho*(\Omega_X^1))\), where \(\rho*(\Omega_X^1)\) is the vector bundle obtained from \(\Omega_X^1\) via the rational representation ρ.     

Semisimilarity for matrices over a division ring

Two square matrices A and B over a ring R are semisimilar, written A$\text{?}$B, if YAX=B and XBY=A for some (possibly rectangular) matrices X, Y over R. We show that if A and B have the same dimension, and if the ring is a division ring $\text{D}$, then A$\text{?}$B if and only if A² is similar to B² and rank(A^k)=rank(B^k), k=1,2,…