On the structure of endomorphisms of projective modules

This paper proposes a method of representing a deterministic stand-alone output automaton by a minimal characteristic polynomial over a finite field. It is shown that defining various automaton transition and output functions allows using the algorithm developed to build the sets of minimal polynomials, different in their powers. Estimates of the relevant sets are given here. We have identified the relation between a characteristic minimal polynomial and an ergodic stochastic matrix defining the sequence developed by a stand-alone automaton. It is shown that the minimal degree of a characteristic polynomial depends linearly on the power of the automaton state set and on the accuracy of representing the elements of a given limiting vector of a stochastic matrix.

Comultiplication Modules over Noncommutative Rings

Black box linear algebra algorithms treat matrices as black boxes that can be applied to input vectors but whose elements cannot be accessed or modified. Since they avoid the fill-in of non-zero entries that occurs with elimination based approaches, black box algorithms provide an efficient approach to computing with sparse matrices and can be used to compute matrix invariants such as minimal polynomial, characteristic polynomial, rank, determinant, and Frobenius normal form for very large sparse matrices over finite fields. Many of these algorithms are probabilistic and often require preconditioning to ensure certain properties hold with high probability. To guarantee a high probability of success it is often necessary that the underlying field contain a large number of elements. Consequently, when computing over small fields, such as GF(2), which is the case for many important applications, these algorithms may require computation over an extension field or repeated computation. This can lead to a significant slow down in practice. Block black box algorithms, such as the block Wiedemann algorithm, which apply black box matrices simultaneously to blocks of input vectors, have been developed for linear system solving. Block algorithms improve performance through better data locality and parallelism and can avoid the need for repeated computation or the use of extension fields needed by approaches without blocking. In this thesis, adaptations of the block Wiedemann algorithm for computing minimal polynomial, characteristic polynomial, rank, determinant, and Frobenius normal form over finite fields, especially small fields, are designed, implemented, analyzed and compared to existing implementations in the LinBox linear algebra library. The key algorithm, LIFs, uses the block Wiedemann algorithm to compute the largest invariant factors of a matrix. A tight probability bound is obtained showing that, with high probability, LIFs correctly computes the r largest invariant factors, even for small fields, using a block size slightly larger than r. An efficient implementation of LIFs requires fast computation of the Smith normal form (SNF) of a polynomial matrix over a finite field. This thesis compares a variety of SNF algorithms and provides an efficient implementation appropriate for use in LIFs. When LIFs is able to compute all of the invariant factors then it can be used to compute minimum polynomial, rank, determinant, characteristic polynomial and Frobenius normal form significantly faster, for matrices over small fields, than existing approaches based on the scalar Wiedemann algorithm with various preconditioners. When there are a large number of invariant factors LIFs cannot be used directly. For rank and determinant computation heuristic preconditioners are used to produce equivalent matrices with a small number of invariant factors for which LIFs can be used. For characterstic polynomial and Frobenius normal form LIFs is used iteratively and for small fields, even when a relatively large block size is needed, the iterated LIFs algorithm outperforms existing approaches. When very large block sizes are needed a new approach for Frobenius form computation based on repeated rank computation is outlined and experiments show it to be promising compared to general black box algorithms for Frobenius normal form.

The object of this paper is to study the relationship between certain projective modules and their endomorphism rings. Specifically, the basic problem is to describe the projective modules whose endomorphism rings are (von Neumann) regular, local semiperfect, or left perfect. Call a projective module regular if every cyclic submodule is a direct summand. Thus a ring is a regular module if it is a regular ring. It is shown that many other equivalent “regularity” conditions characterize regular modules. (For example, every homomorphic image is flat.) Every projective module over a regular ring is regular and a number of examples of regular modules over nonregular rings are given. A structure theorem is obtained: every regular module is isomorphic to a direct sum of principal left ideals. It is shown that the endomorphism ring of a finitely generated regular module is a regular ring. Conversely, over a commutative ring a projective module having a regular endomorphism ring is a regular module. Examples are produced to show that these results are the best possible in the sense that the hypotheses of finite generation and commutativity are needed. An application of these investigations is that a ring $R$ is semisimple with minimum condition if and only if the ring of infinite row matrices over $R$ is a regular ring. Next projective modules having local, semiperfect and left perfect endomorphism rings are studied. It is shown that a projective module has a local endomorphism ring if and only if it is a cyclic module with a unique maximal ideal. More generally, a projective module has a semiperfect endomorphism ring if and only if it is a finite direct sum of modules each of which has a local endomorphism ring.

The group and the minimal polynomial of a graph

Special rank-one perturbations

Let A be the linear transformation on the linear space V in the field P, V λ i be the root subspace corresponding to the characteristic polynomial of the eigenvalue λ i , and W λ i be the root subspace corresponding to the minimum polynomial of λ i . Consider the problem of whether V λ i and W λ i are equal under the condition that the characteristic polynomial of A has the same eigenvalue as the minimum polynomial (see Theorem 1, 2). This article uses the method of mutual inclusion to prove that V λ i = W λ i . Compared to previous studies and proofs, the results of this research can be directly cited in related works. For instance, they can be directly cited in Daoji Meng's book "Introduction to Differential Geometry."

Linear dynamical systems over finite rings

This goal of this chapter is to find finitely many canonical representatives of each similarity class of square matrices with entries in a field and correspondingly of each isomorphism class of linear maps from a finite-dimensional vector space to itself.Section 1 frames the problem in more detail. Section 2 develops the theory of determinants over a commutative ring with identity in order to be able to work easily with characteristic polynomials det(?I — A). The discussion is built around the principle of "permanence of identities," which allows for passage from certain identities with integer coefficients to identities with coefficients in the ring in question.Section 3 introduces the minimal polynomial of a square matrix or linear map. The Cayley-Hamilton Theorem establishes that such a matrix satisfies its characteristic equation, and it follows that the minimal polynomial divides the characteristic polynomial. It is proved that a matrix is similar to a diagonal matrix if and only if its minimal polynomial is the product of distinct factors of degree 1. In combination with the fact that two diagonal matrices are similar if and only if their diagonal entries are permutations of one another, this result solves the canonical-form problem for matrices whose minimal polynomial is the product of distinct factors of degree 1.Section 4 introduces general projection operators from a vector space to itself and relates them to vector-space direct-sum decompositions with finitely many summands. The summands of a directsum decomposition are invariant under a linear map if and only if the linear map commutes with each of the projections associated to the direct-sum decomposition.Section 5 concerns the Primary Decomposition Theorem, whose subject is the operation of a linear map L: V ? V with V finite-dimensional. The statement is that if L has minimal polynomial \( P_1 (\lambda )^{l_1 } \cdots P_k (\lambda )^{l_k } \) with the Pj (?) distinct monic prime, then V has a unique direct-sum decomposition in which the respective summands are the kernels of the linear maps \( P_j (L)^{l_j } \), and moreover the minimal polynomial of the restriction of L to the j th summand is \( P_j (\lambda )^{l_j } \).Sections 6–7 concern Jordan canonical form. For the case that the prime factors of the minimal polynomial of a square matrix all have degree 1, the main theorem gives a canonical form under similarity, saying that a given matrix is similar to one in "Jordan form" and that the Jordan form is completely determined up to permutation of the constituent blocks. The theorem applies to all square matrices if the field is algebraically closed, as is the case for C. The theorem is stated and proved in Section 6, and Section 7 shows how to make computations in two different ways.

This goal of this chapter is to find finitely many canonical representatives of each similarity class of square matrices with entries in a field and correspondingly of each isomorphism class of linear maps from a finite-dimensional vector space to itself. Section 1 frames the problem in more detail. Section 2 develops the theory of determinants over a commutative ring with identity in order to be able to work easily with characteristic polynomials $\det(X I-A)$. The discussion is built around the principle of “permanence of identities,” which allows for passage from certain identities with integer coefficients to identities with coefficients in the ring in question. Section 3 introduces the minimal polynomial of a square matrix or linear map. The Cayley–Hamilton Theorem establishes that such a matrix satisfies its characteristic equation, and it follows that the minimal polynomial divides the characteristic polynomial. It is proved that a matrix is similar to a diagonal matrix if and only if its minimal polynomial is the product of distinct factors of degree 1. In combination with the fact that two diagonal matrices are similar if and only if their diagonal entries are permutations of one another, this result solves the canonical-form problem for matrices whose minimal polynomial is the product of distinct factors of degree 1. Section 4 introduces general projection operators from a vector space to itself and relates them to vector-space direct-sum decompositions with finitely many summands. The summands of a direct-sum decomposition are invariant under a linear map if and only if the linear map commutes with each of the projections associated to the direct-sum decomposition. Section 5 concerns the Primary Decomposition Theorem, whose subject is the operation of a linear map $L:V\to V$ with $V$ finite-dimensional. The statement is that if $L$ has minimal polynomial $P_1(X)^{l_1}\cdots P_k(X)^{l_k}$ with the $P_j(X)$ distinct monic prime, then $V$ has a unique direct-sum decomposition in which the respective summands are the kernels of the linear maps $P_j(L)^{l_j}$, and moreover the minimal polynomial of the restriction of $L$ to the $j^\mathrm{th}$ summand is $P_j(X)^{l_j}$. Sections 6–7 concern Jordan canonical form. For the case that the prime factors of the minimal polynomial of a square matrix all have degree 1, the main theorem gives a canonical form under similarity, saying that a given matrix is similar to one in “Jordan form” and that the Jordan form is completely determined up to permutation of the constituent blocks. The theorem applies to all square matrices if the field is algebraically closed, as is the case for $\mathbb C$. The theorem is stated and proved in Section 6, and Section 7 shows how to make computations in two different ways.

Let R R be a ring and E = E ( R R ) E = E(R_R) its injective envelope. We show that if every simple right R R -module embeds in R R R_R and every cyclic submodule of E R E_R is essentially embeddable in a projective module, then R R R_R has finite essential socle. As a consequence, we prove that if each finitely generated right R R -module is essentially embeddable in a projective module, then R R is a quasi-Frobenius ring. We also obtain several other applications and, among them: a) we answer affirmatively a question of Al-Huzali, Jain, and López-Permouth, by showing that a right CEP ring (i.e., a ring R R such that every cyclic right module is essentially embeddable in a projective module) is always right artinian; b) we prove that if R R is right FGF (i.e., any finitely generated right R R -module embeds in a free module) and right CS, then R R is quasi-Frobenius.

A totally-real polynomial in Z [ x ] \mathbb {Z}[x] with zeros α 1 ⩽ α 2 ⩽ ⋯ ⩽ α d \alpha _1 \leqslant \alpha _2 \leqslant \dots \leqslant \alpha _d has span α d − α 1 \alpha _d - \alpha _1 . Building on the classification of all characteristic polynomials of integer symmetric matrices having small span (span less than 4 4 ), we obtain a classification of small-span polynomials that are the characteristic polynomial of a Hermitian matrix over some quadratic integer ring. Taking quadratic integer rings as our base, we obtain as characteristic polynomials some low-degree small-span polynomials that are not the characteristic (or minimal) polynomial of any integer symmetric matrix.

Linear complexity over [formula omitted] and over [formula omitted] for linear recurring sequences

All rings are commutative with identity and all modules are unital. Let R be a ring and M an R-module. In our recent work [6] we investigated faithful multiplication modules and the properties they have in common with projective modules. In this article, we continue our study and investigate faithful multiplication and locally cyclic projective modules and give several properties for them. If M is either faithful multiplication or locally cyclic projective then M is locally either zero or isomorphic to R. We show that, if M is a faithful multiplication module or a locally cyclic projective module, then for every submodule N of M there exists a unique ideal Γ(N) ⊆ Tr(M) such that N = Γ(N)M. We use this result to show that the structure of submodules of a faithful multplication or locally cyclic projective module and their traces are closely related. We also use the trace of locally cyclic projective modules to study their endomorphisms.

Let LSR 1, LSR2, ..., LSRk and LSR be k+1 linear feedback shift registers with characteristic polynomials f1(x), f2(x), ..., fk(x) and g(x) over \( \mathbb{F}_2 \) and output sequences a 1, a 2, ..., a k and b respectively, where a i=(ai0, ai1, ...), i=1, 2, ..., k, b=(b0, b1, ...). Let \( \mathbb{F}_2^k = \left\{ {\left( {c_1 , c_2 , ..., c_k } \right)\left| {c_k \in \mathbb{F}_2 } \right.} \right\} \) be the k-dimensional space over \( \mathbb{F}_2 \) and γ be an injective map from \( \mathbb{F}_2^k \) into the set {0, 1, 2, ..., n−1}, 2k⩽n, of course. Constructing k-dimensional vector sequence A=(A0, A1, ...) where At=(a1t, a2t, ..., akt), t=0, 1, 2, 3, ... and applying γ to each term of the sequence A, we get the sequence γ(A)=(γ(A0), γ(A1), ...) where γ(At) ∈ {0, 1, ..., n−1}, for all t. Using γ(A) to scramble the output sequence b of LSR, we get the sequence u=(u0, u1, ...) where \( u_t = b_{t + \gamma \left( {A_t } \right)} \), for all t. we call γ a scrambling function and u the Generalized Multiplexed Sequence (generalizing Jenning’s Multiplexed Sequence, see ref, [1]), in brief, GMS. In the present paper, the period, characteristic polynomial, minimum polynomial and translation equivalence properties of the GMS are studied under certain assumptions. Let Ω be the algebraic closure of \( \mathbb{F}_2 \). Throughout this paper, andy algebraic extension of \( \mathbb{F}_2 \) are assumed to be contained in Ω. Let f(x) and g(x) be polynomials over \( \mathbb{F}_2 \) without multiple roots. Let f*g be the monic polynomial whose roots are all the distinct elements of the set S={α·β | α, β∈Ω, f(α)=0, g(β)=0}. It is well known that f*g is polynomial over \( \mathbb{F}_2 \). Let G(f) denote the vector space consisting of all output sequences of LSR with characteristic polynomial f(x).