Elementary Column Research Articles

A FAST COMPUTATION FOR EIGENVALUES OF CIRCULANT MATRICES WITH ARITHMETIC SEQUENCE

In this article, we derive simple formulations of the eigenvalues, determinants, and also the inverse of circulant matrices whose entries in the first row form an arithmetic sequence. The formulation of the determinant and inverse is based on elementary row and column operations transforming the matrix to an equivalent diagonal matrix so that the formulation is obtained easily. Meanwhile, for the eigenvalues formulation, we simplify the known result of formulation for the general circulant matrices by exploiting the properties of the cyclic group induced by the set of all roots of as the set of points in the unit circle in the complex plane, and also by considering the specific property of arithmetic sequence. Then, we construct an algorithm for the eigenvalues formulation. This algorithm shows a better computation compared to the previously known result for the general case of circulant matrices.

A COMPUTATION PERSPECTIVE FOR THE EIGENVALUES OF CIRCULANT MATRICES INVOLVING GEOMETRIC PROGRESSION

In this article, the eigenvalues and inverse of circulant matrices with entries in the first row having the form of a geometric sequence are formulated explicitly in a simple form in one theorem. The method for deriving the formulation of the determinant and inverse is simply using elementary row or column operations. For the eigenvalues, the known formulation of the previous results is simplified by considering the specialty of the sequence and using cyclic group properties of unit circles in the complex plane. Then, the algorithm of eigenvalues formulation is constructed, and it shows as a better computation method.

Numerical algorithms for the determinants of opposite-bordered and singly-bordered tridiagonal matrices

A recursive algorithm for the determinant evaluation of general opposite-bordered tridiagonal matrices has been proposed by Jia et al. (J Comput Appl Math 290:423–432, 2015). Since the algorithm is a symbolic algorithm, it never suffers from breakdown. However, it may be time-consuming when many symbolic names emerge during the symbolic computation. In this paper, without using symbolic computation, first we present a novel breakdown-free numerical algorithm for computing the determinant of an n-by-n opposite-bordered tridiagonal matrix, which does not require any extra memory storage for the implementation. Then, we present a cost-efficient algorithm for the determinants of opposite-bordered tridiagonal matrices based on the use of the combination of an elementary column operation and Sylvester’s determinant identity. Furthermore, we provide some numerical results with simulations in Matlab implementation in order to demonstrate the accuracy and efficiency of the proposed algorithms, and their competitiveness with other existing algorithms. The corresponding results in this paper can be readily obtained for computing the determinants of singly-bordered tridiagonal matrices.

DOA Estimation without Source Number for Cyber-Physical Interactions

In this paper, a direction of arrival (DOA) estimator is proposed to improve the cyber-physical interactions, which is based on the second-order statistics without a priori knowledge of the source number. The impact of noise will firstly be eliminated. Then the relationship between the processed covariance matrix and the steering matrix is studied. By applying the elementary column transformation, an oblique projector will be designed without the source number. At last, a rooting method will be adopted to estimate the DOAs with the constructed projector. Simulation results show that the proposed method performs as well as other methods, which requires that the source number must be known.

The WST-decomposition for partial matrices

A partial matrix over a field F is a matrix whose entries are either an element of F or an indeterminate, and each indeterminate only appears once. A completion is a matrix obtained by giving values to each of the indeterminates. We can decompose any partial matrix, through elementary row operations and column permutations, into a block matrix of the form [W⁎⁎0S⁎00T] where W is wide (has more columns than rows), S is square, T is tall (has more rows than columns), and with these three blocks having at least one completion with full rank. And importantly, W, S and T are unique up to elementary row operations and column permutations whenever S is required to be as large as possible. When this is the case, [W⁎⁎0S⁎00T] will be called a WST-decomposition. As an application of this decomposition, we will see that the maximal rank over all completions of the original partial matrix is rows(W)+rows(S)+cols(T). In fact, we introduce the WST-decomposition for a broader class of matrices: the ACI-matrices.

Computation of weighted Moore–Penrose inverse through Gauss–Jordan elimination on bordered matrices

In this paper, two new algorithms for computing the Weighted Moore–Penrose inverse AM,N† of a general matrix A for weights M and N which are based on elementary row and column operations on two appropriate block partitioned matrices are introduced and investigated. The computational complexity of the introduced two algorithms is analyzed in detail. These two algorithms proposed in this paper are always faster than those in Sheng and Chen (2013) and Ji (2014), respectively, by comparing their computational complexities. In the end, an example is presented to demonstrate the two new algorithms.

Fast joint sorting and reduction algorithm for MIMO SIC detection

A fast joint sorting and reduction (FJSR) algorithm is proposed, which involves elementary column operations only, to assist the successive interference cancellation (SIC) detection for multiple-input multiple-output (MIMO) systems. Simulation results show that the FJSR-aided SIC detector achieves good error performance especially when the system scales up, yet requires only a few extra basis updates over the optimal-ordered SIC detector regardless of the system size. This suggests that FJSR is particularly advantageous to small-to-medium scale MIMO systems with strict delay constraints..

English

English

Analysis of the descriptor Roesser model with the use of the Drazin inverse

Abstract A method of analysis for a class of descriptor 2D discrete-time linear systems described by the Roesser model with a regular pencil is proposed. The method is based on the transformation of the model to a special form with the use of elementary row and column operations and on the application of a Drazin inverse of matrices to handle the model. The method is illustrated with a numerical example

Generalized synchronization of arbitrary-dimensional chaotic systems

In this paper, the generalized synchronization of nonlinear chaotic systems is investigated. Based upon reversibility of the Jacobian matrix and matrix theory, we carry out a series of elementary column transformations to Jacobian matrix. And then, according to Lyapunov function theory, sufficient conditions for generalized synchronization between drive system and response system with different dimensions are obtained. Numerical simulations are also given to demonstrate the effectiveness of the proposed scheme.

Design of regular positive and stable descriptor systems via state-feedbacks for descriptor continuous-time linear systems with singular pencils

Abstract A new method is proposed of design of regular positive and asymptotically stable descriptor systems by the use of state-feedbacks for descriptor continuous-time linear systems with singular pencils. The method is based on the reduction of the descriptor system by elementary row and column operations to special form. A procedure for the design of the state-feedbacks gain matrix is presented and illustrated by a numerical example

Convolutional cylinder-type block-circulant cycle codes

In this paper‎, ‎we consider a class of column-weight two‎ ‎quasi-cyclic low-density parity check codes in which the girth can be large enough‎, ‎as an‎ ‎arbitrary multiple of 8‎. ‎Then we devote a convolutional form to these codes‎, ‎such that‎ ‎their generator matrix can be obtained by elementary row and‎ ‎column operations on the parity-check matrix‎. ‎Finally‎, ‎we show that the free distance of the convolutional codes is equal to the minimum distance of‎ ‎their block counterparts‎.

Innovation based on Gaussian elimination to compute generalized inverse [formula omitted

In this paper, we execute elementary row and column operations on the partitioned matrix (GAGGG0) into ((Is000)00−AT,S(2))to compute generalized inverse AT,S(2) of a given complex matrix A, where G is a matrix such that R(G)=T and N(G)=S. The total number of multiplications and divisions operations is T(m,n,s)=2mn2+4m−s−12ns+(m−s)ns+mns and the upper bound of T(m,n,s) is less than 6mn2−32n3−12n2 when n≤m. A numerical example is shown to illustrate that this method is correct.

Positive fractional continuous-time linear systems with singular pencils

Positive fractional continuous-time linear systems with singular pencils A method for checking the positivity and finding the solution to the positive fractional descriptor continuous-time linear systems with singular pencils is proposed. The method is based on elementary row and column operations of the fractional descriptor systems to equivalent standard systems with some algebraic constraints on state variables and inputs. Necessary and sufficient conditions for the positivity of the fractional descriptor systems are established.

Checking of the positivity of descriptor linear systems with singular pencils

Checking of the positivity of descriptor linear systems with singular pencilsA method for checking of the positivity of descriptor continuous-time and discrete-time linear systems with singular pencil is proposed. The method is based on elementary row and column operations on the matrices of descriptor systems. Necessary and sufficient conditions for the positivity of the descriptor systems are established.

One-peak posets with positive quadratic Tits form, their mesh translation quivers of roots, and programming in Maple and Python

By applying linear algebra and computer algebra tools we study finite posets with positive quadratic Tits form. Our study is motivated by applications of matrix representations of posets in representation theory, where a matrix representation of a partially ordered set T={p1,…,pn}, with a partial order ⪯, means a block matrix M=[M1|M2|…|Mn] (over a field K) of size d*×(d1,…,dn) determined up to all elementary row transformations, elementary column transformations within each of the substrips M1,M2,…,Mn, and additions of linear combinations of columns of Mi to columns of Mj, if pi≺pj. Drozd [10] proves that T has only a finite number of direct-sum-indecomposable representations if and only if its quadratic Tits form q(x1,…,xn,x*)=x12+⋯+xn2+x*2+∑pi≺pjxixj-x*(x1+⋯+xn) is weakly positive (i.e., q(a1,…,an,a*)>0, for all non-zero vectors (a1,…,an,a*) with integral non-negative coefficients). In this case, there exists an indecomposable representation M of size d*×(d1,…,dn) if and only if (d1,…,dn,d*) is a root of q, i.e., q(d1,…,dn,d*)=1. Bondarenko and Stepochkina [8] give a list of posets T with positive Tits form consisting of four infinite series and 108 posets, up to duality. In the paper, we construct this list in an alternative way by applying computational algorithms implemented in Maple and Python. Moreover, given any poset T of the list, we show that: (a) the Coxeter polynomial coxI(t) of the poset I=T∪{*}, obtained from T by adding a unique maximal element * (called a peak), is a Coxeter polynomial FΔI(t) of a simply-laced Dynkin diagram ΔI∈{An,Dn,E6,E7,E8}and (b) the set of Φ-orbits of the set Rq of integral roots of q admits a Φ-mesh translation quiver structure Γ(Rq,Φ) of a cylinder shape, where Φ is the Coxeter transformation of I=T∪{*} in the sense of Drozd [10].

Restrictions as stabilizers

Two GF ( q ) -representations of a matroid are projectively equivalent if one can be obtained from the other by elementary row operations or column scaling. If, in addition, we allow field automorphisms, then the representations are equivalent. A matroid stabilizes its single-element extensions over GF ( q ) , if no two GF ( q ) -representations of the matroid can be extended to two projectively inequivalent GF ( q ) -representations of its extension. We prove that if a 3-connected GF ( q ) -representable matroid stabilizes its single-element extensions, then it stabilizes any GF ( q ) -representable matroid that contains it as a restriction. As a result, a GF ( q ) -representable matroid with a line containing at least q points is stabilized by that line and a GF ( q ) -representable matroid with a plane containing at least 2 q points, but no line containing at least q points, is stabilized by that plane.

Surgical treatment of dystonia

Surgical treatment of dystonia has experienced a tremendous change over the past decade. Whilst selective peripheral denervation is reserved for cervical dystonia refractory to botulinum toxin injections, deep brain stimulation (DBS) of the pallidum has gained a wide scope and presents an elementary column in the treatment of medically refractory patients, nowadays. There is consensus that idiopathic generalized, cervical and segmental dystonia are good indications for DBS, although there is still a paucity of studies providing high-level data according to EBM criteria. Efficacy is maintained on longterm. Several other forms of primary dystonia are still under investigation but it appears that patients with Meige syndrome and myoclonus-dystonia gain also marked benefit. Study of the outcome in secondary dystonia disorders is more complex, in general, but patients with tardive dystonia gain similar improvement than patients with idiopathic dystonia. Overall, the risk profile of pallidal DBS is quite low, and it has been shown to be cognitively safe. The effect of pallidal DBS on non-dystonic extremities has not received much attention, albeit there are hints for a pro-akinetic mechanism. Several questions remain to be solved including optimal programming of stimulation settings, battery drain with high stimulation energies and the elucidation of the mechanisms of DBS in dystonia.

Iteration, not induction

The main purpose of this note is to present and justify proof via iteration as an intuitive, creative and empowering method that is often available and preferable as an alternative to proofs via either mathematical induction or the well-ordering principle. The method of iteration depends only on the fact that any strictly decreasing sequence of positive integers must terminate in finitely many steps. Four examples are presented using the method of iteration. These examples concern the factorization of integers or polynomials into products of primes, the calculation of quotients and remainders via the division algorithm (and, hence, of greatest common divisors via the Euclidean Algorithm) and the rank of matrices via elementary row and column operations. We also justify the method of proof by iteration by showing that its validity is a logical consequence of the method of mathematical induction. Various parts of this material would be suitable for courses on Precalculus, Discrete Mathematics, Algebra or Matrix Theory/Linear Algebra.

Column and diagonal‐reduction of polynomial matrices by orthogonal transformation

AbstractMany of the applications of polynomial matrices in real world systems require column‐ or diagonally‐reduced polynomial matrices. If a given polynomial matrix is not column‐ or diagonally‐reduced, Callier or Wolowich algorithms, which use unimodular transformations, can be applied for column‐ or diagonal‐reduction, respectively, as a pre‐processing step in the applications. However, Callier and Wolowich algorithms may be unstable, from a numerical viewpoint, because they use elementary column and row operations. The purpose of this paper is to present sufficient conditions for existence of a constant orthogonal transformation of the given polynomial matrix so that it becomes column‐ or diagonally‐reduced. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society