A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. The symmetric sign pattern matrices that require unique inertia have recently been characterized. The purpose of this paper is to more generall...A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. The symmetric sign pattern matrices that require unique inertia have recently been characterized. The purpose of this paper is to more generally investigate the inertia sets of symmetric sign pattern matrices. In particular, nonnegative tri-diagonal sign patterns and the square sign pattern with all + entries are examined. An algorithm is given for generating nonnegative real symmetric Toeplitz matrices with zero diagonal of orders n≥3 which have exactly two negative eigenvalues. The inertia set of the square pattern with all + off-diagonal entries and zero diagonal entries is then analyzed. The types of inertias which can be in the inertia set of any sign pattern are also obtained in the paper. Specifically, certain compatibility and consecutiveness properties are established.展开更多
This paper is concerned with root localization of a complex polynomial with respect to the unit circle in the more general case. The classical Schur-Cohn-Fujiwara theorem converts the inertia problem of a polynomial t...This paper is concerned with root localization of a complex polynomial with respect to the unit circle in the more general case. The classical Schur-Cohn-Fujiwara theorem converts the inertia problem of a polynomial to that of an appropriate Hermitian matrix under the condition that the associated Bezout matrix is nonsingular. To complete it, we discuss an extended version of the Schur-Cohn-Fujiwara theorem to the singular case of that Bezout matrix. Our method is mainly based on a perturbation technique for a Bezout matrix. As an application of these results and methods, we further obtain an explicit formula for the number of roots of a polynomial located on the upper half part of the unit circle as well.展开更多
We derive the solvability conditions and an expression of the general solution to the system of matrix equations A 1X=C1 , A2Y=C2 , YB2=D2 , Y=Y*, A3Z=C3 , ZB3=D3 , Z=Z*, B4X+(B4X)+C4YC4*+D4ZD4*=A4 . Moreover, we inve...We derive the solvability conditions and an expression of the general solution to the system of matrix equations A 1X=C1 , A2Y=C2 , YB2=D2 , Y=Y*, A3Z=C3 , ZB3=D3 , Z=Z*, B4X+(B4X)+C4YC4*+D4ZD4*=A4 . Moreover, we investigate the maximal and minimal ranks and inertias of Y and Z in the above system of matrix equations. As a special case of the results, we solve the problem proposed in Farid, Moslehian, Wang and Wu's recent paper (Farid F O, Moslehian M S, Wang Q W, et al. On the Hermitian solutions to a system of adjointable operator equations. Linear Algebra Appl, 2012, 437: 1854-1891).展开更多
为提升低惯量电力系统的小干扰稳定性,给出保障适应新能源发展的合理惯量裕度,并明确惯量分布对于互联系统小干扰稳定性的影响规律。研究了惯量分布对系统区域间振荡模式的影响规律,构建了面向区域间振荡的电力系统小扰动惯量域(Small S...为提升低惯量电力系统的小干扰稳定性,给出保障适应新能源发展的合理惯量裕度,并明确惯量分布对于互联系统小干扰稳定性的影响规律。研究了惯量分布对系统区域间振荡模式的影响规律,构建了面向区域间振荡的电力系统小扰动惯量域(Small Signal Inertia Region,SSIR)。首先,基于部分惯量中心(Part of the Center of Inertia,PCOI)等值方法和多项式Leverrier解法,构建了等值两机系统区间振荡模式阻尼比解析式。在此基础上,依据临界阻尼比构建了计及小干扰稳定约束的惯量域。进而,对区域互联系统惯量分布和小干扰稳定性之间的联系进行解析。最后,对惯量域进行全面分析,并在新能源系统中进行了初步验证。仿真结果表明,所提出的计算方法能够实现电力系统小扰动惯量域的准确、快速构建,增强低惯量电力系统小干扰稳定性的评估和监控能力,并为新能源惯量域的构建提供理论指导。展开更多
In this paper, we present a fast and fraction free procedure for computing the inertia of Bezout matrix and we can determine the numbers of different real roots and different pairs of conjugate complex roots of a pol...In this paper, we present a fast and fraction free procedure for computing the inertia of Bezout matrix and we can determine the numbers of different real roots and different pairs of conjugate complex roots of a polynomial equation with integer coefficients quickly based on this result.展开更多
Generally, the least-squares problem can be solved by the normal equation. Based on the projection theorem, we propose a direct method to investigate the maximal and minimal ranks and inertias of the least-squares sol...Generally, the least-squares problem can be solved by the normal equation. Based on the projection theorem, we propose a direct method to investigate the maximal and minimal ranks and inertias of the least-squares solutions of matrix equation AXB = C under Hermitian constraint, and the corresponding formulas for calculating the rank and inertia are derived.展开更多
In this paper, the inertia of a symmetric Z-matrix is studied, and bounds of the number of its positive eigenvaues are obtained. Also the interlacing theorem for Schur complement of a symmetric Z-matrix is established...In this paper, the inertia of a symmetric Z-matrix is studied, and bounds of the number of its positive eigenvaues are obtained. Also the interlacing theorem for Schur complement of a symmetric Z-matrix is established, which can be considered as a generalization Cauchy interlacing theorem in some extent.展开更多
文摘A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. The symmetric sign pattern matrices that require unique inertia have recently been characterized. The purpose of this paper is to more generally investigate the inertia sets of symmetric sign pattern matrices. In particular, nonnegative tri-diagonal sign patterns and the square sign pattern with all + entries are examined. An algorithm is given for generating nonnegative real symmetric Toeplitz matrices with zero diagonal of orders n≥3 which have exactly two negative eigenvalues. The inertia set of the square pattern with all + off-diagonal entries and zero diagonal entries is then analyzed. The types of inertias which can be in the inertia set of any sign pattern are also obtained in the paper. Specifically, certain compatibility and consecutiveness properties are established.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Nos. 11071017, 11271045） and the Program for New Century Excellent Talents in University.
文摘This paper is concerned with root localization of a complex polynomial with respect to the unit circle in the more general case. The classical Schur-Cohn-Fujiwara theorem converts the inertia problem of a polynomial to that of an appropriate Hermitian matrix under the condition that the associated Bezout matrix is nonsingular. To complete it, we discuss an extended version of the Schur-Cohn-Fujiwara theorem to the singular case of that Bezout matrix. Our method is mainly based on a perturbation technique for a Bezout matrix. As an application of these results and methods, we further obtain an explicit formula for the number of roots of a polynomial located on the upper half part of the unit circle as well.
基金National Natural Science Foundation of China (Grant No. 11171205)Natural Science Foundation of Shanghai (Grant No. 11ZR1412500)+2 种基金the Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20093108110001)Shanghai Leading Academic Discipline Project (Grant No. J50101)Innovation Foundation of Shanghai University (Grant No. SHUCX120109)
文摘We derive the solvability conditions and an expression of the general solution to the system of matrix equations A 1X=C1 , A2Y=C2 , YB2=D2 , Y=Y*, A3Z=C3 , ZB3=D3 , Z=Z*, B4X+(B4X)+C4YC4*+D4ZD4*=A4 . Moreover, we investigate the maximal and minimal ranks and inertias of Y and Z in the above system of matrix equations. As a special case of the results, we solve the problem proposed in Farid, Moslehian, Wang and Wu's recent paper (Farid F O, Moslehian M S, Wang Q W, et al. On the Hermitian solutions to a system of adjointable operator equations. Linear Algebra Appl, 2012, 437: 1854-1891).
文摘为提升低惯量电力系统的小干扰稳定性,给出保障适应新能源发展的合理惯量裕度,并明确惯量分布对于互联系统小干扰稳定性的影响规律。研究了惯量分布对系统区域间振荡模式的影响规律,构建了面向区域间振荡的电力系统小扰动惯量域(Small Signal Inertia Region,SSIR)。首先,基于部分惯量中心(Part of the Center of Inertia,PCOI)等值方法和多项式Leverrier解法,构建了等值两机系统区间振荡模式阻尼比解析式。在此基础上,依据临界阻尼比构建了计及小干扰稳定约束的惯量域。进而,对区域互联系统惯量分布和小干扰稳定性之间的联系进行解析。最后,对惯量域进行全面分析,并在新能源系统中进行了初步验证。仿真结果表明,所提出的计算方法能够实现电力系统小扰动惯量域的准确、快速构建,增强低惯量电力系统小干扰稳定性的评估和监控能力,并为新能源惯量域的构建提供理论指导。
文摘In this paper, we present a fast and fraction free procedure for computing the inertia of Bezout matrix and we can determine the numbers of different real roots and different pairs of conjugate complex roots of a polynomial equation with integer coefficients quickly based on this result.
基金Supported by the Science Foundation Project of Tianshui Normal University(TSA1315)
文摘Generally, the least-squares problem can be solved by the normal equation. Based on the projection theorem, we propose a direct method to investigate the maximal and minimal ranks and inertias of the least-squares solutions of matrix equation AXB = C under Hermitian constraint, and the corresponding formulas for calculating the rank and inertia are derived.
文摘In this paper, the inertia of a symmetric Z-matrix is studied, and bounds of the number of its positive eigenvaues are obtained. Also the interlacing theorem for Schur complement of a symmetric Z-matrix is established, which can be considered as a generalization Cauchy interlacing theorem in some extent.
