Inverse Eigenvalue Problem for Generalized Arrow-Like Matrices

This paper researches the following inverse eigenvalue problem for arrow-like matrices. Give two characteristic pairs, get a generalized arrow-like matrix, let the two characteristic pairs are the characteristic pairs of this generalized arrow-like matrix. The expression and an algorithm of the solution of the problem is given, and a numerical example is provided.

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

is gained by deleting the kth row and the kth column （k=1,2,...,n） from Tn.We put for-ward an inverse eigenvalue problem to be that:If we don’t know the matrix T1,n,but weknow all eigenvalues of matrix T1,k-1,all eigenvalues of matrix Tk+1,k,and all eigenvaluesof matrix T1,n could we construct the matrix T1,n.Let μ1,μ2,…,μk-1,μk,μk+1,…,μn-1,

REVERSE ORDER LAW FOR GROUP INVERSE OF PRODUCT OF MATRICES ON SKEW FIELDS

Let Kn×n be the set of all n×n matrices and Krn×n the set {A∈Kn×n|rankA=r} on askew field K. Zhuang [1] denotes by A# the group inverse of A∈Kn×n which is the solu-tion of the euqations:AXA=A, XAX=X, AX=AX.

THE DRAZIN INVERSE OF MATRICES OF QUATERNIONS

Let H be the real quaternion field,C and R be the complex and real field respectively.Clearly R（?）C（?）H. Let Hm×n denote the set of all m×n matrices over H.If A=(ars)∈Hm×n,then there exist A1 and A2∈Cm×n such that A=A1+A2j.Let AC denote the complexrepresentation of A,that is the 2m×2n complex matrix Ac=（（A1/A2）（-A2/A1））（see[1,2]）.We denote by AD the Drazin inverse of A∈Hm×n which is the unique solution of the e-

Solubility Existence of Inverse Eigenvalue Problem for a Class of Singular Hermitian Matrices

In this article,we discuss singular Hermitian matrices of rank greater or equal to four for an inverse eigenvalue problem.Specifically,we look into how to generate n by n singular Hermitian matrices of ranks four and five from a prescribed spectrum.Numerical examples are presented in each case to illustrate these scenarios.It was established that given a prescribed spectral datum and it multiplies,then the solubility of the inverse eigenvalue problem for n by n singular Hermitian matrices of rank r exists.

New Approach for the Inversion of Structured Matrices via Newton’s Iteration

Newton’s iteration is a fundamental tool for numerical solutions of systems of equations. The well-known iteration ?rapidly refines a crude initial approximation X0?to the inverse of a general nonsingular matrix. In this paper, we will extend and apply this method to n× n?structured matrices M?, in which matrix multiplication has a lower computational cost. These matrices can be represented by their short generators which allow faster computations based on the displacement operators tool. However, the length of the generators is tend to grow and the iterations do not preserve matrix structure. So, the main goal is to control the growth of the length of the short displacement generators so that we can operate with matrices of low rank and carry out the computations much faster. In order to achieve our goal, we will compress the computed approximations to the inverse to yield a superfast algorithm. We will describe two different compression techniques based on the SVD and substitution and we will analyze these approaches. Our main algorithm can be applied to more general classes of structured matrices.

BEST APPROXIMATION BY NORMAL MATRICES WITH SPECTRAL CONSTRAINTS

考虑在给定谱约束和Ｆｒｏｂｅｎｉｕｓ范数意义下用正规矩阵最佳逼近一个给定复方阵的问题。给出了这个问题可解的充分必要条件，提出了求解这个问题的一个数值算法，并给出了一个数值例子。

The Explicit Expressions of Level- k(r1 ,r2 ,… ,rk) Circulant Matrices and Their Some Properties

In this paper,we give the explicit expressions of level k (r 1,r 2,…,r k) circulant matrices of order n 1n 2…n k,and the explicit expressions for the eigenvalues,the determinants and the inverse matrices of the kind level k (r 1,r 2,…,r k) circulant matrices are derived,and it is also proved that the sort of matrices are diagonalizable.

The Explicit Expressions of Level-k Circulant Matrices of Type(n_1,n_2.…,n_k) and Their Some Properties

In this paper, we give the explicit expressions of level-k circulant matrices of type (n1,n2,…nk) and of order n1n2…nk,and the explicit expressions for the eigenvalues,the determinants and the inverse matrices of the kind level-k circulant matrices are derived,and it is also proved that the sort matrices are unitarily diagonalizable.

对称五对角矩阵加箭型矩阵的广义逆谱问题

研究了一类对称五对角矩阵加箭型矩阵的广义逆谱问题,将该矩阵所有主子阵的极端特征值作为其特征数据,利用几何学上的圆锥曲线、五对角矩阵及箭型矩阵的相关性质,重构此类特殊箭状矩阵。最后给出了该问题解的充分条件以及问题构造的算法与实例,验证了结果的准确性。

两类非对称双箭型矩阵的广义逆谱问题

针对两类非对称双箭型矩阵的广义逆谱问题,本文先将两类矩阵的两组特征对作为其特征数据,然后利用矩阵元素间具有的函数关系、线性关系及箭型矩阵的相关性质,将两类矩阵的逆谱问题转换为求解线性方程组的问题,进而实现了两类矩阵的重构。本文给出了该问题有唯一解的充分必要条件以及问题构造的算法,并通过相应数值实例验证了所得结果。

关于“Key agreement scheme based on generalised inverses of matrices”的一点补充

本文指出E.Dawson和Chuan-Kuan Wu在1997年给出的密钥协定方案须取m≠n,并给出一种遵守任何情况下(包含m=n情况)的修正方案.

THE SOLVABILITY CONDITIONS FOR THE INVERSE PROBLEM OF MATRICES POSITIVE SEMIDEFINITE ON A SUBSPACE

This paper studies the following two problems: Problem I. Given X, B is-an-element-of R(n x m), find A is-an-element-of P(s,n), such that AX = B, where Ps, n = {A is-an-element-of SR(n x n)\x(T) Ax greater-than-or-equal-to 0, for-all S(T) x = 0, for given S is-an-element-of R(p)n x p}. Problem II. Given A* is-an-element-of R(n x n), find A is-an-element-of S(E), such that \\A*-A\\ = inf(A is-an-element-of S(E) \\A*-A\\ where S(E) denotes the solution set of Problem I. The necessary and sufficient conditions for the solvability of Problem I, the expression of the general solution of Problem I and the solution of Problem II are given for two cases. For the general case, the equivalent form of conditions for the solvability of Problem I is given.

In-Place Matrix Inversion by Modified Gauss-Jordan Algorithm

The classical Gauss-Jordan method for matrix inversion involves augmenting the matrix with a unit matrix and requires a workspace twice as large as the original matrix as well as computational operations to be performed on both the original and the unit matrix. A modified version of the method for performing the inversion without explicitly generating the unit matrix by replicating its functionality within the original matrix space for more efficient utilization of computational resources is presented in this article. Although the algorithm described here picks the pivots solely from the diagonal which, therefore, may not contain a zero, it did not pose any problem for the author because he used it to invert structural stiffness matrices which met this requirement. Techniques such as row/column swapping to handle off-diagonal pivots are also applicable to this method but are beyond the scope of this article.

Generalized Inverse Eigenvalue Problem for (P,Q)-Conjugate Matrices and the Associated Approximation Problem

In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the least residual problem of the above generalized inverse eigenvalue problem is studied by using the canonical correlation decomposition（CCD）.The solutions to these problems are derived.Some numerical examples are given to illustrate the main results.

Inverse-preserving Linear Maps Between Spaces of Matrices over Fields

Suppose F is a field different from F2, the field with two elements. Let Mn（F） and Sn（F） be the space of n × n full matrices and the space of n ×n symmetric matrices over F, respectively. For any G1, G2 ∈ {Sn（F）, Mn（F）}, we say that a linear map f from G1 to G2 is inverse-preserving if f（X）^-1 = f（X^-1） for every invertible X ∈ G1. Let L （G1, G2） denote the set of all inverse-preserving linear maps from G1 to G2. In this paper the sets .L（Sn（F）,Mn（F））, L（Sn（F）,Sn（F））, L （Mn（F）,Mn（F）） and L（Mn （F）, Sn （F）） are characterized.

STRUCTURES OF CIRCULANT INVERSE M-MATRICES

In this paper, we present a useful result on the structures of circulant inverse Mmatrices. It is shown that if the n × n nonnegative circulant matrix A = Circ[c0, c1,… , c（n- 1）] is not a positive matrix and not equal to c0I, then A is an inverse M-matrix if and only if there exists a positive integer k, which is a proper factor of n, such that cjk 〉 0 for j=0,1…, [n-k/k], the other ci are zero and Circ[co, ck,… , c（n-k）] is an inverse M-matrix. The result is then extended to the so-called generalized circulant inverse M-matrices.

矩阵之和Drazin逆的表示及其应用

通过将矩阵之和转化为矩阵之积的思想,利用矩阵Drazin逆的定义、性质,将和矩阵Drazin逆问题转化为三角分块矩阵的Drazin逆问题,给出了在一定条件下和矩阵Drazin逆新的表示,进而给出分块矩阵在更弱条件下Drazin逆的表示,最后通过算例来验证结果的科学性。

Generalized Drazin spectrum of upper triangular matrices in Banach algebras

Let A be a Banach algebra with unit e and a,b,c∈A,Mc=(a c 0 b)∈M_(2)(A).The concepts of left and right generalized Drazin invertible of elements in a Banach algebra are proposed.A generalized Drazin spectrum of is defined byσ_(gD)(α)={λ∈C:α-λe is not generalized Drazin invertible}.It is shown thatσ_(gD)(a)∪σ_(gD)(b)=σ_(gD)(M_(C))∪W_(2),where W_(g) is a union of certain holes σ_(gD) and W_(g)■σ_(gD)(a)∩σ_(gD)(b),or more finely.In addition,some properties of generalized Drazin spectrum of elements in a Banach algebra are studied.

Banach代数中反三角矩阵的p群逆

研究了几类反三角矩阵的p群可逆性,利用p群逆的定义对其进行验证,得到了这几类反三角矩阵p群逆的表示.并以这些矩阵为基础探究某些特殊矩阵p群逆的表示,根据两个积为0的p群可逆元素的和的p群逆公式,计算得到了结果.