Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Polynomial n × n matrices A(x) and B(x) over a field \mathbbF \mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over \mathbbF \mathbb{F} and an invertible n × n matrix Q(x) over \mathbbF \mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A _0x - A ₁, where A ₀ and A ₁ are n × n matrices over \mathbbF \mathbb{F} , and A ₀ is nonsingular.     

Products of idempotent matrices

For some years it has been known that every singular square matrix over an arbitrary field F is a product of idempotent matrices over F. This paper quantifies that result to some extent. Main result: for every field F and every pair (n,k) of positive integers, an n×n matrix S over F is a product of k idempotent matrices over F iff rank(I ? S)?k· nullity S. The proof of the “if” part involves only elementary matrix operations and may thus be regarded as constructive. Corollary: (for every field F and every positive integer n) each singular n×n matrix over F is a product of n idempotent matrices over F, and there is a singular n×n matrix over F which is not a product of n ? 1 idempotent matrices.     

Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.     

Conley's spectral sequence via the sweeping algorithm

In this article we consider a spectral sequence (Er,dr) associated to a filtered Morse-Conley chain complex (C,Δ), where Δ is a connection matrix. The underlying motivation is to understand connection matrices under continuation. We show how the spectral sequence is completely determined by a family of connection matrices. This family is obtained by a sweeping algorithm for Δ over fields F as well as over Z. This algorithm constructs a sequence of similar matrices Δ⁰=Δ,Δ¹,… , where each matrix is related to the others via a change-of-basis matrix. Each matrix Δr over F (resp., over Z) determines the vector space (resp., Z-module) Er and the differential dr. We also prove the integrality of the final matrix ΔR produced by the sweeping algorithm over Z which is quite surprising, mainly because the intermediate matrices in the process may not have this property. Several other properties of the change-of-basis matrices as well as the intermediate matrices Δr are obtained.     

Flows on graphs applied to diagonal similarity and diagonal equivalence for matrices

Three equivalence relations are considered on the set of n × n matrices with elements in F₀, an abelian group with absorbing zero adjoined. They are the relations of diagonal similarity, diagonal equivalence, and restricted diagonal equivalence. These relations are usually considered for matrices with elements in a field. But only multiplication is involved. Thus our formulation in terms of an abelian group with o is natural. Moreover, if F is chosen to be an additive group, diagonal similarity is characterized in terms of flows on the pattern graph of the matrices and diagonal equivalence in terms of flows on the bipartie graph of the matrices. For restricted diagonal equivalence a pseudo-diagonal of the graph must also be considered. When no pseudo-diagonal is present, the divisibility properties of the group F play a role. We show that the three relations are characterized by cyclic, polygonal, and pseudo-diagonal products for multiplicative F. Thus, our method of reducing propositions concerning the three equivalence relations to propositions concerning flows on graphs, provides a unified approach to problems previously considered independently, and yields some n, w or improved results. Our consideration of cycles rather than circuits eliminates certain restrictions (e.g., the complete reducibility of the matrices) which have previously been imposed. Our results extend theorems in Engel and Schneider [5], where however the group F is permitted to be non-commutative.     

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.     

Essentially doubly stochastic matrices

In this paper we extend the general theory of essentially doubly stochastic (e.d.s.) matrices begun in earlier papers in this series. We complete the investigation in one direction by characterizing all of the algebra isomorphisms between the algebra of e.d.s. matrices of order n over a field F,E_n(F), and the total algebra of matrices of order n - 1over F,M_n-1(F) We then develop some of the theory when Fis a field with an involution. We show that for any e,f§Fof norm 1,e≠f every e.d.s. matrix in E_n(F) is a unique e.d.s. sum of an e.d.s. e-hermitian matrix and an e.d.s. f-hermitian matrix in E_n(F) Next, we completely determine the cases for which there exists an above-mentioned matrix algebra isomorphism preserving adjoints. Finally, we consider cogredience in E_n(F) and show that when such an adjoint-preserving isomorphism exists and char M_n(F) two e.d.s. e-hermitian matrices which are cogredient in M_n(F) are also cogredient in E_n(F). Using this result, we obtain simple canonical forms for cogredience of e.d.s. e-hermitian matrices in E_n(F) when Fsatisfies special conditions. This ncludes the e.d.s. skew-symmetric matrices, where the involution is trivial and E = -1.     

Adequate equivalence relations and Pontryagin products

Let A be an abelian variety over a field k. We consider CH₀(A) as a ring under Pontryagin product and relate powers of the ideal I⊆CH₀(A) of degree zero elements to powers of the algebraic equivalence relation. We also consider a filtration F⁰⊇F¹⊇… on the Chow groups of varieties of the form T×_kA (defined using Pontryagin products on A×_kA considered as an A-scheme via projection on the first factor) and prove that F^r coincides with the r-fold product (F¹)^*r as adequate equivalence relations on the category of all such varieties.     

Types of superregular matrices and the number of n‐arcs and complete n‐arcs in PG (r,q)

Based on the classification of superregular matrices, the numbers of non‐equivalent n‐arcs and complete n‐arcs in PG(r, q) are determined (i) for 4 ≤ q ≤ 19, 2 ≤ r ≤ q ? 2 and arbitrary n, (ii) for 23 ≤ q ≤ 32, r = 2 and n ≥ q ? 8<$>. The equivalence classes over both PGL (k, q) and PΓL(k, q) are considered throughout the examinations and computations. For the classification, an n‐arc is represented by the systematic generator matrix of the corresponding MDS code, without the identity matrix part of it. A rectangular matrix like this is superregular, i.e., it has only non‐singular square submatrices. Four types of superregular matrices are studied and the non‐equivalent superregular matrices of different types are stored in databases. Some particular results on t(r, q) and m′(r, q)—the smallest and the second largest size for complete arcs in PG(r, q)—are also reported, stating that m′(2, 31) = 22, m′(2, 32) = 24, t(3, 23) = 10, and m′(3, 23) = 16. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 363–390, 2006     

On Certain Power-Associative,Lie-Admissible Subalgebras of Matrix Algebras

The theory of functional identities is applied to the classification of the third-power-associative products * which can be defined on certain Lie subalgebras A of the matrix algebra M _n (F) over a field F such that x * y − y * x = xy − yx for all x, y ∈ A, where xy denotes the usual associative product in M _n (F) and A is the matrix algebra itself, a Lie ideal, a one-sided ideal, the Lie algebra of skew elements, or the algebra of upper triangular matrices. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 13, Algebra, 2004.     

Generalized inverses of matrices over a finite field

For a given m × n matrix A of rank r over a finite field F, the number of generalized inverses, of reflexive generalized inverses, of normalized generalized inverses, and of pseudoinverses of A are determined by elementary methods. The more difficult problem of determining which m × n matrices A of rank r over F have normalized generalized inverses and which have pseudoinverses is solved. Moreover, the number of such matrices which possess normalized generalized inverses and the number which possess pseudoinverses are found.     

A criterion for generating full matrix algebras

Let M_n(F) be the algebra of n×n matrices over a field F, and let A∈M_n(F) have characteristic polynomial c(x)=p₁(x)p₂(x)?p_r(x) where p₁(x),…,p_r(x) are distinct and irreducible in F[x]. Let X be a subalgebra of M_n(F) containing A. Under a mild hypothesis on the p_i(x), we find a necessary and sufficient condition for X to be M_n(F).     

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.     

The number of equivalence classes of nondegenerate bilinear and sesquilinear forms over a finite field

Generating functions in the form of infinite products are given for the number of equivalence classes of nondegenerate sesquilinear forms of rank n over GF(q²) and for the number of equivalence (or congruence) classes of nondegenerate bilinear forms of rank n over GF(q).     

Explicit functorial correspondences for level zero representations of p-adic linear groups

Let F be a non-Archimedean local field and D a central F-division algebra of dimension n², n?1. We consider first the irreducible smooth representations of D^× trivial on 1-units, and second the indecomposable, n-dimensional, semisimple, Weil-Deligne representations of F which are trivial on wild inertia. The sets of equivalence classes of these two sorts of representations are in canonical (functorial) bijection via the composition of the Jacquet-Langlands correspondence and the Langlands correspondence. They are also in canonical bijection via explicit parametrizations in terms of tame admissible pairs. This paper gives the relation between these two bijections. It is based on analysis of the discrete series of the general linear group GLn(F) in terms of a classification by extended simple types.     

Counting and constructing orthogonal circulants

If F is an arbitrary finite field and T is an n × n orthogonal matrix with entries in F then one may ask how to find all the orthogonal matrices belonging to the algebra F[T] and one may want to know the cardinality of this group. We present here a means of constructing this group of orthogonal matrices given the complete factorization of the minimal polynomial of T over F. As a corollary of this construction scheme we give an explicit formula for the number of n × n orthogonal circulant matrices over GF(p^l) and a similar formula for symmetric circulants. These generalize results of MacWilliams, J. Combinatorial Theory10 (1971), 1–17.     

Classification of pairs consisting of a linear and a semilinear map

L. Kronecker has found normal forms for pairs (A, B) of m-by-n matrices over a field F when the admissible transformations are of the type (A, B)→(SAT, SBT), where S and T are invertible matrices over F. For the details about these normal forms we refer to Gantmacher's book on matrices [5, Chapter XII]. See also Dickson's paper [3]. We treat here the following more general problem: Find the normal forms for pairs (A, B) of m-by-n matrices over a division ring D if the admissible transformations are of the type (A, B)→(SAT, SBJ(T)) where J is an automorphism of D. It is surprising that these normal forms (see Theorem 1) are as simple as in the classical case treated by Kronecker. The special case D=C, J=conjugation is essentially equivalent to the recent problem of Dlab and Ringel [4]. This is explained thoroughly in Sec. 6. We conclude with two open problems.     

An extension of Birkhoff’s theorem with an application to determinants

Let M_n(F) denote the algebra of n×n matrices over the field F of complex, or real, numbers. Given a self-adjoint involution J∈M_n(C), that is, J=J^*,J²=I, let us consider Cⁿ endowed with the indefinite inner product [,] induced by J and defined by [x,y]?〈Jx,y〉,x,y∈Cⁿ. Assuming that (r,n-r), 0?r?n, is the inertia of J, without loss of generality we may assume J=diag(j₁,?,j_n)=I_r⊕-I_n-r. For T=(|t_ik|²)∈M_n(R), the matrices of the form T=(|t_ik|²j_ij_k), with all line sums equal to 1, are called J-doubly stochastic matrices. In the particular case r∈{0,n}, these matrices reduce to doubly stochastic matrices, that is, non-negative real matrices with all line sums equal to 1. A generalization of Birkhoff’s theorem on doubly stochastic matrices is obtained for J-doubly stochastic matrices and an application to determinants is presented.     

Witt equivalence of central simple algebras with involution

The notion of Witt equivalence of central simple algebras with involution is introduced. It is shown that the standard invariants, i.e. the discriminant, the signature and the Clifford algebra, depend only on the Witt class of the algebra with involution. For a given filedF the tensor product is used to construct a semigroup\(\tilde S\left( F \right)\) and this semigroup is shown to have properties analogous to the multiplicative properties of the Witt ring of quadratic forms overF.     

Intersection problems and integral equivalence

Let A and B be matrices over a principal ideal domain, Π. Necessary conditions, involving the invariant factors of A and B, are given for B to be a submatrix of A or a principal submatrix of A.If a given nonnegative integral matrix, B, is the intersection matrix of a pair of families of subsets of an n-set, and n is the smallest integer for which this is true, we say that the content of B is n. In that event, B is a submatrix of K(n), the intersection matrix of all subsets of an n-set. More refined results are obtained in certain cases by considering S(n, k, l), the intersection matrix of the k-subsets of an n-set versus its l-subsets. The invariant factors of K(n) and S(n, k, l) are calculated and it is shown how this information may be used to get lower bounds for the content of B. In the more widely studied symmetric version of the content problem, B must be a principal submatrix of K(n) or, possibly, S(n, k) = S(n, k, k). In this case, the invariant factors of K(n) ? xI or S(n, k) ? xI also provide relevant information.