SYMMETRIC POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATIONS (AX,XB)=(C,D) AND AXB=C

The symmetric positive definite solutions of matrix equations (AX,XB)=(C,D) and AXB=C are considered in this paper. Necessary and sufficient conditions for the matrix equations to have symmetric positive definite solutions are derived using the singular value and the generalized singular value decompositions. The expressions for the general symmetric positive definite solutions are given when certain conditions hold.

Serum matrix metalloproteinase 3 in detecting remitting seronegative symmetrical synovitis with pitting edema syndrome: A case report

We report a case of remitting seronegative symmetrical synovitis with pitting edema(RS3 PE) syndrome in a 71-year-old woman. She referred to our hospital with finger stiffness, edema of both hands and feet, pain of bilateral shoulder, wrist, metacarpophalangeal, proximal interphalangeal, and ankle joints. Rheumatoid factor was negative, human leukocyte antigen-B7 antigen was positive. Moreover, matrix metalloproteinase 3(MMP-3) was high. She was diagnosed with RS3 PE syndrome, and treatment with prednisolone(15 mg/d) was started. One week after prednisolone treatment initiation, CRP decreased to negative, and joint pain was almost completely resolved. However, hand stiffness persisted, and MMP-3 level was still high. Thus, prednisolone dose was increased to 20 mg/d, and the stiffness resolved. Twenty days after treatment initiation, MMP-3 was normalized. MMP-3 was more indicative of RS3 PE syndrome symptoms than CRP. Thus, MMP-3 seems to be more sensitive to RS3 PE syndrome symptoms.

The General and Centro(Kewsy) Symmetric Solutions to a System of Matrix Equations over an Arbitrary Skew Field

Necessary and sufficient conditions are given for the existence of the general solution, the centrosymmetric solution, and the centroskewsymmetric solution to a system of linear matrix equations over an arbitrary skew field. The representations of such the solutions of the system are also derived.

On the Centrosymmetric and Centroskewsymmetric Solutions to a Matrix Equation over a Central Algebra

Let Ω be a finite dimensional central algebra and chart Ω≠2 .The matrix equation AXB-CXD=E over Ω is considered.Necessary and sufficient conditions for the existence of centro(skew)symmetric solutions of the matrix equation are given.As a particular case ,the matrix equation X-AXB=C over Ω is also considered.

THE SYMMETRIC AND SYMMETRIC POSITIVE SEMIDEFINITE SOLUTIONS OF LINEAR MATRIX EQUATION——B^TXB = D ON LINEAR MANIFOLDS

This paper discusses the solutions of the linear matrix equation BT X B=Don some linear manifolds.Some necessary and sufficient conditions for the existenceof the solution and the expression of the general solution are given.And also someoptimal approximation solutions are discussed.

Voltage and Current Mode Vector Analyses of Correction Procedure Application to Clarke’s Matrix—Symmetrical Three-Phase Cases

Clarke’s matrix has been applied as a phase-mode transformation matrix to three-phase transmission lines substituting the eigenvector matrices. Considering symmetrical untransposed three-phase lines, an actual symmetrical three-phase line on untransposed conditions is associated with Clarke’s matrix for error and frequency scan analyses in this paper. Error analyses are calculated for the eigenvalue diagonal elements obtained from Clarke’s matrix. The eigenvalue off-diagonal elements from the Clarke’s matrix application are compared to the correspondent exact eigenvalues. Based on the characteristic impedance and propagation function values, the frequency scan analyses show that there are great differences between the Clarke’s matrix results and the exact ones, considering frequency values from 10 kHz to 1 MHz. A correction procedure is applied obtaining two new transformation matrices. These matrices lead to good approximated results when compared to the exact ones. With the correction procedure applied to Clarke’s matrix, the relative values of the eigenvalue matrix off-diagonal element obtained from Clarke’s matrix are decreased while the frequency scan results are improved. The steps of correction procedure application are detailed, investigating the influence of each step on the obtained two new phase-mode transformation matrices.

Derivative of a Determinant with Respect to an Eigenvalue in the <i>LDU</i>Decomposition of a Non-Symmetric Matrix

We demonstrate that, when computing the LDU decomposition (a typical example of a direct solution method), it is possible to obtain the derivative of a determinant with respect to an eigenvalue of a non-symmetric matrix. Our proposed method augments an LDU decomposition program with an additional routine to obtain a program for easily evaluating the derivative of a determinant with respect to an eigenvalue. The proposed method follows simply from the process of solving simultaneous linear equations and is particularly effective for band matrices, for which memory requirements are significantly reduced compared to those for dense matrices. We discuss the theory underlying our proposed method and present detailed algorithms for implementing it.

On Eigenvalues Locations of Symmetric Matrix Families

It is proved that the set of all symmetric real matrices of order n with eigenvalues lying in the interval(α, β), denoted by Sn(α,β), is convex in Rn×n. With this result, some known results on positive(negative) definiteness, and Hurwitz(Shur) stability, as well as the aperiodic property of polytopes of symmetric matrices are generalized, and a series of insightful necessary and sufficient conditions for some general set of symmetric matrices contained in Sn(α,β) are presented,which are directly available for analysis of the positive(negative) definiteness, Hurwitz(Shur) stability and the aperiodic property of a wide class of sets of symmetric matrices.

Finding the Maximal Eigenpair for a Large, Dense, Symmetric Matrix based on Mufa Chen's Algorithm

A hybrid method is presented for determining maximal eigenvalue and its eigenvector(called eigenpair)of a large,dense,symmetric matrix.Many problems require finding only a small part of the eigenpairs,and some require only the maximal one.In a series of papers,efficient algorithms have been developed by Mufa Chen for computing the maximal eigenpairs of tridiagonal matrices with positive off-diagonal elements.The key idea is to explicitly construet effective initial guess of the maximal eigenpair and then to employ a self-closed iterative algorithm.In this paper we will extend Mufa Chen's algorithm to find maximal eigenpair for a large scale,dense,symmetric matrix.Our strategy is to first convert the underlying matrix into the tridiagonal form by using similarity transformations.We then handle the cases that prevent us from applying Chen's algorithm directly,e.g.,the cases with zero or negative super-or sub-diagonal elements.Serval numerical experiments are carried out to demonstrate the efficiency of the proposed hybrid method.

Derivative of a Determinant with Respect to an Eigenvalue in the Modified Cholesky Decomposition of a Symmetric Matrix, with Applications to Nonlinear Analysis

In this paper, we obtain a formula for the derivative of a determinant with respect to an eigenvalue in the modified Cholesky decomposition of a symmetric matrix, a characteristic example of a direct solution method in computational linear algebra. We apply our proposed formula to a technique used in nonlinear finite-element methods and discuss methods for determining singular points, such as bifurcation points and limit points. In our proposed method, the increment in arc length (or other relevant quantities) may be determined automatically, allowing a reduction in the number of basic parameters. The method is particularly effective for banded matrices, which allow a significant reduction in memory requirements as compared to dense matrices. We discuss the theoretical foundations of our proposed method, present algorithms and programs that implement it, and conduct numerical experiments to investigate its effectiveness.

Security Implementation in WSN with Symmetric and Matrix Mapping on Asymmetric ECC Cryptographic Techniques

As the wireless sensor networks are easily deployable, the volume of sensor applications has been increased widely in various fields of military and commercial areas. In order to attain security on the data exchanged over the network, a hybrid cryptographic mechanism which includes both symmetric and asymmetric cryptographic functions is used. The public key cryptographic ECC security implementation in this paper performs a matrix mapping of data’s at the points on the elliptical curve, which are further encoded using the private symmetric cipher cryptographic algorithm. This security enhancement with the hybrid mechanism of ECC and symmetric cipher cryptographic scheme achieves efficiency in energy conservation of about 7% and 4% compared to the asymmetric and symmetric cipher security implementations in WSN.

AN IMPROVEMENT ON THE QL ALGORITHM FOR SYMMETRIC TRIDIAGONAL MATRICES

This paper establishes an improvement on the QL algorithm for a symmetric tridiagonal matrix T so that we can work out the eigenvalues of T faster. Meanwhile, the new algorithm don’t worsen the stability and precision of the former algorithm.

AN ESTIMATE OF EIGENVALUES IN THE SYMMETRIC QR ALGORITHM

The estimate of the eigenvalues is given when the off-diagonal elements in symmetric tridiagonal matrix are replaced by zero. The result can be applied to QR or QL algorithm. It is a generalization of Jiang’ s result in 1987. This estimate is sharper than Hager’s result in 1982 and could not

QL Method for Symmetric Tridiagonal Matrices

QL(QR) method is an efficient method to find eigenvalues of a matrix. Especially we use QL(QR) method to find eigenvalues of a symmetric tridiagonal matrix. In this case it only costs O(n2) flops, to find all eigenvalues. So it is one of the most efficient method for symmetric tridiagonal matrices. Many experts have researched it. Even the method is mature, it still has many problems need to be researched. We put forward five problems here. They are: (1) Convergence and convergence rate; (2) The convergence of diagonal elements; (3) Shift designed to produce the eigenvalues in monotone order; (4) QL algorithm with multi-shift; (5) Error bound. We intoduce our works on these problems, some of them were published and some are new.

A Class of Constrained Inverse Eigenproblem and Associated Approximation Problem for Symmetric Reflexive Matrices

Let S∈Rn×n be a symmetric and nontrival involution matrix. We say that A∈E R n×n is a symmetric reflexive matrix if AT = A and SAS = A. Let S R r n×n(S)={A|A= AT,A = SAS, A∈Rn×n}. This paper discusses the following two problems. The first one is as follows. Given Z∈Rn×m (m < n),∧= diag(λ1,...,λm)∈Rm×m, andα,β∈R withα<β. Find a subset (?)(Z,∧,α,β) of SRrn×n(S) such that AZ = Z∧holds for any A∈(?)(Z,∧,α,β) and the remaining eigenvaluesλm+1 ,...,λn of A are located in the interval [α,β], Moreover, for a given B∈Rn×n, the second problem is to find AB∈(?)(Z,∧,α,β) such that where ||.|| is the Frobenius norm. Using the properties of symmetric reflexive matrices, the two problems are essentially decomposed into the same kind of subproblems for two real symmetric matrices with smaller dimensions, and then the expressions of the general solution for the two problems are derived.

The κ-point Exponent Set of Central Symmetric Primitive Matrices

Let G =（Y, E） be a primitive digraph. The vertex exponent of G at a vertex v E V, denoted by expG（v）, is the least integer p such that there is a v → u walk of length p for each u E V. We choose to order the vertices of G in such a way that expG（v1） ≤ expG（v2） ≤... ≤expG（vn）. Then expG（vk） is called the k-point exponent of G and is denoted by expG（k）, 1 〈 k 〈 n. We define the k-point exponent set E（n, k） ：= {expG（k）｜G= G（A） with A E CSP（n）｜, where CSP（n） is the set of all n × n central symmetric primitive matrices and G（A） is the associated graph of the matrix A. In this paper, we describe E（n, k） for all n, k with 1 〈 k 〈 n except n ≡ l（mod 2） and 1 〈 k 〈 n - 4. We also characterize the extremal graphs when k = 1.

Higher-Order Duality for Minimax Fractional Type Programming Involving Symmetric Matrices

Convexity and generalized convexity play important roles in optimization theory. With the development of programming problem, there has been a growing interest in the higher-order dual problem and a lot of related generalized convexities are given. In this paper, we give the convexity of (F, α ,p ,d ,b , φ )β vector-pseudo- quasi-Type I and formulate a higher-order duality for minimax fractional type programming involving symmetric matrices, and give the weak, strong and strict converse duality theorems under the condition of higher-order (F, α ,p ,d ,b , φ )β vector-pseudoquasi-Type I.

The Exponent Set of Central Symmetric Primitive Matrices

This paper first establishes a distance inequality of the associated dia-graph of a central symmetric primitive matrix, then characters the exponent set of central symmetric primitive matrices, and proves that the exponent set of central symmetric primitive matrices of order n is {1, 2,…, n - 1}. There is no gap in it.

Jordan Semi-Triple Multiplicative Maps on the Symmetric Matrices

In this paper, we show that if an injective map on symmetric matrices Sn(C) satisfies then for all , where f is an injective homomorphism on C, S is a complex orthogonal matrix and Af is the image of A under f applied entrywise.