Rank-k-preservers and preservers of sets of ranks

Let T be a linear transformation on the set of m × n matrices with entries in an algebraically closed field. If T maps the set of all matrices whose rank is k into itself, and if$\text{n?}\text{3}\text{k}\text{2}$, then the rank of A is the rank of T(A) for every m × n matrix.     

On the Schur multiplier norm of matrices

The Schur product of two n×n complex matrices A=(a_ij), B=(b_ij) is defined by A°B=(a_ijb_ij. By a result of Schur [2], the algebra of n×n matrices with Schur product and the usual addition is a commutative Banach algebra under the operator norm (the norm of the operator defined on $\text{C}$ⁿ by the matrix). For a fixed matrix A, the norm of the operator B?A°B on this Banach algebra is called the Schur multiplier norm of A, and is denoted by ∥A∥^m. It is proved here that $\text{∥A∥=∥U}{}^1\text{AU∥}{}_{\mathrm{m}}$ for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥A∥_m=∥A∥ iff there exist two permutations P, Q, a p×p (1?p?n) unitary U, an (n?p)×(n?p)1 contraction C, and a nonnegative number λ such that

$\text{A=λP}\text{U}\text{0}\text{0}\text{C}\text{Q;}$

and this is so iff $\text{∥A°}\text{A}\text{?}\text{∥=∥A∥}{}^2$, where ā is the matrix obtained by taking entrywise conjugates of A.     

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).     

The weight distribution of irreducible cyclic codes with block lengths n1((q1−1)N)

We study the weight distribution of irreducible cyclic (n, k) codeswith block lengths n = n₁((q¹ ? 1)/N), where N|q ? 1, gcd(n₁,N) = 1, and gcd(l,N) = 1. We present the weight enumerator polynomial, A(z), when k = n₁l, k = (n₁ ? 1)l, and k = 2l. We also show how to find A(z) in general by studying the generator matrix of an (n₁, m) linear code, $\text{V}{}^1{}_{\mathrm{d}}$ over GF(q^d) where d = gcd (ord_n1(q), l). Specifically we study A(z) when $\text{V}{}^1{}_{\mathrm{d}}$ is a maximum distance separable code, a maximal shiftregister code, and a semiprimitive code. We tabulate some numbers A_μ which completely determine the weight distributionof any irreducible cyclic (n₁(2¹ ? 1), k) code over GF(2) for all n₁ ? 17.     

Conditions under which the order of two non-negative Hermitian matrices is preserved when taking integral powers

Let A and B be two n×n non-negative matrices. We write A ? B iff

$\text{u}{}^1\text{(A ? B)u ? 0}$

for all column vectors u in Cⁿ. Here u¹ is the conjugate transpose of u. In this paper are stated equivalent conditions under which A^k ? B^k for all natural numbers k. The result is then generalized to Hermitian operators in a Hilbert space.     

Duality between some linear preserver problems. II. Isometries with respect to c-special norms and matrices with fixed singular values

Let $\text{F}{}_{\mathrm{m\times n}}$ (m?n) denote the linear space of all m × n complex or real matrices according as $\text{F}$=$\text{C}$ or $\text{R}$. Let c=(c₁,…,c_m)≠0 be such that c₁???c_m?0. The c-spectral norm of a matrix A?$\text{F}$_m×n is the quantity

$\text{6A6}{}_{\mathrm{c}}\text{∑}\text{i=I}\text{m}\text{c}{}_{\mathrm{i}}\text{σ}{}_{\mathrm{i}}\text{(A)}$

. where σ₁(A)???σ_m(A) are the singular values of A. Let d=(d₁,…,d_m)≠0, where d₁???d_m?0. We consider the linear isometries between the normed spaces $\text{(}\text{F}{}_{\mathrm{m\times }}{}_{\mathrm{n}}\text{,∥·∥}{}_{\mathrm{<mtext>c</mtext>}}\text{)}$ and $\text{(}\text{F}{}_{\mathrm{m\times }}{}_{\mathrm{n}}\text{,∥·∥}{}_{\mathrm{d}}\text{)}$, and prove that they are dual transformations of the linear operators which map $\text{L}$(d) onto $\text{L}$(c), where

$\text{L}\text{(c)= {X?}\text{F}{}_{\mathrm{m\times n}}\text{:X}\text{has singular values}\text{c}{}_1\text{,…,c}{}_{\mathrm{m}}\text{}}$

.     

Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

For a sequence A = {A_k} of finite subsets of N we introduce: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{n}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$, where A(m) is the number of subsets A_k ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation $\text{a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b)}$ constitutes a finite semi-group N^∪ (semi-group N^∩) (group $\text{N}{}^1$). For N^∪, N^∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)^?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {A_kr}, where A_kr | A_kr+1 for r = 1, 2, …. In Section 6 we consider for $\text{N∪, N∩, N1}$ analogues of Rohrbach inequality: $\text{2n}\text{? g(n) ? 2}\text{n}$, where g(n) = min k over the subsets {a₁ < … < a_k} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = a_i + a_j.Pour une série A = {A_k} de sous-ensembles finis de N on introduit les densités: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{m}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$ où A(m) est le nombre d'ensembles A_k ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations $\text{a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b}$, un semi-groupe fini N^∪, N^∩ ou un groupe N¹ respectivement. Pour N^∪, N^∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)^?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {A_kr}, où A_kr|A_kr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N^∪, $\text{N}{}^{\mathrm{\cap }}\text{, N}{}^1$ les analogues de l'inégalité de Rohrbach: $\text{2n}\text{? g(n) ? 2}\text{n}$, où g(n) = min k pour les ensembles {a₁ < … < a_k} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = a_i + a_j.     

Matricial logarithmic derivatives

If θ is a norm on Cⁿ, then the mapping $\text{A→}\text{lim}{}_{\mathrm{h\downarrow 0}}\text{6I+hA6}{}_{\mathrm{\theta }}\text{?1}\text{/h}$ from M_n(C) (=C^{n × n}) into R is called the logarithmic derivative induced by the vector norm θ. In this paper we generalize this concept to a mapping γ from M_n(C) into M_k(R), where k ? n. Denoting by α(B) the spectral abscissa of a square matrix B (the largest of the real parts of the eigenvalues), we show, in particular, that α(A) ?α(γ(A)). As a byproduct we obtain simple sufficient conditions for the stability of a matrix.     

Involutory matrices over finite commutative rings

An n × n matrix A is called involutory iff A²=I_n, where I_n is the n × n identity matrix. This paper is concerned with involutory matrices over an arbitrary finite commutative ring R with identity and with the similarity relation among such matrices. In particular the authors seek a canonical set $\text{C}$ with respect to similarity for the n × n involutory matrices over R—i.e., a set $\text{C}$ of n × n involutory matrices over R with the property that each n × n involutory matrix over R is similar to exactly on matrix in $\text{C}$. Because of the structure of finite commutative rings and because of previous research, they are able to restrict their attention to finite local rings of characteristic a power of 2, and although their main result does not completely specify a canonical set $\text{C}$ for such a ring, it does solve the problem for a special class of rings and shows that a solution to the general case necessarily contains a solution to the classically unsolved problem of simultaneously bringing a sequence A₁,…,A_v of (not necessarily involutory) matrices over a finite field of characteristic 2 to canonical form (using the same similarity transformation on each A_i). (More generally, the authors observe that a theory of similarity fot matrices over an arbitrary local ring, such as the well-known rational canonical theory for matrices over a field, necessarily implies a solution to the simultaneous canonical form problem for matrices over a field.) In a final section they apply their results to find a canonical set for the involutory matrices over the ring of integers modulo 2^m and using this canonical set they are able to obtain a formula for the number of n × n involutory matrices over this ring.     

Algebraic integers and distributions on the N-torus

Let D(T_N) be the class of real-valued, infinitely differentiable, periodic functions, and let D′(T_N) be the class of real-valued distributions on T_N having D(T_N) as its test functions. Define A(T_N) ? D(T_N) as follows: S is in A(T_N) if there is a constant K such that $\text{¦ S^(m)¦ ? K}$ for all m, and, furthermore, lim_{min(|m1|,…,|mN|)←∞}|S^{^}(m)|=0. For $\text{0 < ξ <}\text{1}\text{2}$, let C(ξ) be the familiar Cantor set with constant ratio of dissection ξ, constructed on the interval [?π, π). The following result is established: A necessary and sufficient condition thatC(ξ₁) × … × C(ξ_N)be a set of uniqueness for the classA(T_N)is that eachξ_j^?1be an$\text{J}$number forj = 1,…, N.     

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.     

Scalar polynomial functions on the n×n matrices over a finite field

Let F=GF(q) denote the finite field of order q, and let $\text{?(x)?F[x]}$. Then f(x) defines, via substitution, a function from F_n×n, the n×n matrices over F, to itself. Any function $\text{?:F}{}_{\mathrm{n\times n}}\text{→ F}{}_{\mathrm{n\times n}}$ which can be represented by a polynomialf(x)?F[x] is called a scalar polynomial function on F_n×n. After first determining the number of scalar polynomial functions on F_n×n, the authors find necessary and sufficient conditions on a polynomial $\text{?(x) ? F[x]}$ in order that it defines a permutation of (i) $\text{D}$_n, the diagonalizable matrices in F_n×n, (ii)$\text{R}$_n, the matrices in F_n×n all of whose roots are in F, and (iii) the matric ring F_n×n itself. The results for (i) and (ii) are valid for an arbitrary field F.     

Retarded equations on the sphere induced by linear equations

Using a Poincaré compactification, the linear homogeneous system of delay equations {x = Ax(t ? 1) (A is an n × n real matrix) induces a delay system π(A) on the sphere Sⁿ. The points at infinity belong to an invariant submanifold S^{n ? 1} of Sⁿ. For an open and dense set of 2 × 2 matrices A with distinct eigenvalues, the system π(A) has only hyperbolic critical points (including the critical points at infinity). For an open and dense set of 2 × 2matrices $\text{A}$ with complex eigenvalues, the nonwandering set at infinity is the union of an odd number of hyperbolic periodic orbits; if $\text{(}\text{det}\text{A}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{<}\text{3π}\text{2}$, the restriction of $\text{π(}\text{A}\text{)}$ to S¹ is Morse-Smale. For n = 1 there exist periodic orbits of period 4 provided that $\text{?A >}\text{π}\text{2}$ and Hopf bifurcation of a center occurs for ?A near $\text{(}\text{π}\text{2}\text{) + 2kπ, k ? Z}$.     

Duality in hypercomplex function theory

Let $\text{A}$ be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e₁,…, e_n}, and e₀ be the identity of $\text{A}$. Furthermore, let M_k(Ω;$\text{A}$) be the set of $\text{A}$-valued functions defined in an open subset Ω of R^m+1 (1 ? m ? n) which satisfy D^kf = 0 in Ω, where D is the generalized Cauchy-Riemann operator $\text{D = ∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{m}}\text{e}{}_{\mathrm{i}}\text{(}\text{?}\text{?x}{}_{\mathrm{i}}\text{)}$ and k? N. The aim of this paper is to characterize the dual and bidual of M_k(Ω;$\text{A}$). It is proved that, if M_k(Ω;$\text{A}$) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits M_k(Ω;$\text{A}$) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized.     

On matrices having equal corresponding principal minors

Let A, B be n × n matrices with entries in a field F. We say A and B satisfy property $\text{D}$ if B or B^t is diagonally similar to A. It is clear that if A and B satisfy property $\text{D}$, then they have equal corresponding principal minors, of all orders. The question is to what extent the converse is true. There are examples which show the converse is not always true. We modify the problem slightly and give conditions on a matrix A which guarantee that if B is any matrix which has the same principal minors as A, then A and B will satisfy property $\text{D}$. These conditions on A are formulated in terms of ranks of certain submatrices of A and the concept of irreducibility.     

A note on polynomial matrix functions over a finite field

Let F_n denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=A_Nx^N+ ?+ A₁x+A₀?F_n[x]. A function $\text{?:F}{}_{\mathrm{n}}\text{→F}{}_{\mathrm{n}}$ is called a right polynomial function iff there exists an A(x)?F_n[x] such that $\text{?(B)=A}{}_{\mathrm{N}}\text{B}{}^{\mathrm{N}}\text{+?+A}{}_1\text{B+ A}{}_0$ for every B?F_n. This paper obtains unique representations for and determines the number of right polynomial functions.     

3×3 Orthostochastic matrices and the convexity of generalized numerical ranges

Let $\text{U}$₃ be the set of all 3 × 3 unitary matrices, and let A and B be two 3 × 3 complex nor?al matrices. In this note, the authors first give a necessary and sufficient condition for a 3 × 3 doubly stochastic matrix to be orthostochastic and then use this result to consider the structure of the sets $\text{W}$ (A) = {Diag UAU¹ : U ∈ $\text{U}$₃} and W(A,B) = {Tr UAU¹B: U ∈ $\text{U}$₃}, where ¹ denotes the transpose conjugate.     

Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function

The permanent function is used to determine geometrical properties of the set $\text{Ω}{}_{\mathrm{n}}$ of all n × n nonnegative doubly stochastic matrices. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$, then $\text{F}$ corresponds to an n × n (0, 1)-matrix A, where the permanent of A is the number of vertices of $\text{F}$. If A is fully indecomposable, then the dimension of $\text{F}$ equals σ(A) ? 2n + 1, where σ(A) is the number of 1's in A. The only two-dimensional faces of $\text{Ω}{}_{\mathrm{n}}$ are triangles and rectangles. For n ? 6, $\text{Ω}{}_{\mathrm{n}}$ has four types of three-dimensional faces. The facets of the faces of $\text{Ω}{}_{\mathrm{n}}$ are characterized. Faces of $\text{Ω}{}_{\mathrm{n}}$ which are simplices are determined. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$ which is two-neighborly but not a simplex, then $\text{F}$ has dimension 4 and six vertices. All k-dimensional faces with k + 2 vertices are determined. The maximum number of vertices of a k-dimensional face is 2^k. All k-dimensional faces with at least 2^k?1 + 1 vertices are determined.