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四元数体上重行列式的性质及其应用 总被引:18,自引:1,他引:17
本文得到了四元数体上重行列式的一些基本不等式,给出了矩阵为正定自共轭阵时,行列式与重行列式的显式关系,同时也给出了文[1]中行列式与文[5]中行列式两种定义的关系。提出了四元数体上广义正定自共轭阵的概念,并获得了这类阵的基本性质. 相似文献
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八元数矩阵的行列式及其性质 总被引:1,自引:0,他引:1
赋范的可除代数只有四种:实数R,复数C,四元数日和八元数O.由于八元数关于乘法非交换且非结合,如何对八元数矩阵定义行列式并使其具有较好的运算性质变得非常困难.最近,李兴民和黎丽根据"八元数自共轭矩阵的行列式应为实数"这一数学与物理上的需求,通过选择几个八元数乘积的次序和结合方式,首次给出了八元数行列式的定义.但是,与实数、复数以及四元数的相应的情形比较,如此定义的行列式,其所具备的运算性质较少.本文给出了一种新的八元数行列式的定义,它们具备了尽可能多的运算性质,同时使得"八元数自共轭矩阵的行列式为实数"不证自明. 相似文献
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本文主要研究了交换半环上矩阵乘积行列式的性质。讨论了行列式的乘积与乘积的行列式间的关系,并进一步给出了伴随阵的乘积与乘积的伴随阵之间的关系。 相似文献
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在谢邦杰教授的特征值与行列式定义下,本文定义了自共轭四元数矩阵的惯性,得到了分块的自共轭四元数矩阵的含有广义 Schur 补的惯性公式与行列式公式,并将 Carlso-Haynsworth-Markhan 行列式不等式推广到四元数体上。 相似文献
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本文的日的在于改进已有的两个复矩阵的行列式的上界,以更精细的两个Hermitian正定矩阵和的行列式为基本工具.利用得到的相关一无二次不等式描述的行列式之间的关系,给出了两个复矩阵和的行列式新上界,作为心用可改进华罗庚行列式不等式的上界. 相似文献
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It is shown that the invertibility of a Toeplitz matrix can be determined through the solvability of two standard equations. The inverse matrix can be denoted as a sum of products of circulant matrices and upper triangular Toeplitz matrices. The stability of the inversion formula for a Toeplitz matrix is also considered. 相似文献
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The paper is concerned with the problem of inverting block matrices to which the well-known Frobenius— Schur formulas are not applicable. These can be square matrices with four noninvertible square or rectangular blocks as well as square or rectangular matrices with two blocks. With regard to rectangular matrices, the results obtained are a further step in the development of the canonization method, which is used for solving arbitrary matrix equations. 相似文献
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Gabriela R. Argiroffo Silvia M. Bianchi Graciela L. Nasini 《Mathematical Methods of Operations Research》2008,67(2):245-256
Lehman (Polyhedral combinatorics 1 of DIMACS series in discrete math. and theoretical computer science, pp 101–105, 1990) described some conditions regular
minimally nonideal (mni) matrices must satisfy. Although, there are few results on sufficient conditions for mni matrices.
In most of these results, the covering polyhedron must have a unique fractional extreme point. This condition corresponds
to ask the matrix to be the blocker of a near-ideal matrix, defined by the authors in a previous work (2006). In this paper
we prove that, having the blocker of a near-ideal matrix, only a few very easy conditions have to be checked in order to decide
if the matrix is regular mni. In doing so, we define the class of quasi mni matrices, containing regular mni matrices, and
we find a generalization on the number of integer extreme points adjacent to the fractional extreme point in the covering
polyhedron. We also give a relationship between the covering and stability number of regular mni matrices which allows to
prove when a regular mni matrix can be a proper minor of a quasi mni.
Partially supported by CONICET Grant PIP 2807/2000 (Argentina) and by CNPq/PROSUL Grant 490333/2004-4 (Brasil). 相似文献
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Let be the set of all n × n real matrices which have a positive determinant. We show here that at least 2n ? 1 matrices are needed to “see” each matrix in . Also, any finite subset of can be “seen” from a class of at most 2n ? 1 matrices in . 相似文献
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Marko Huhtanen 《BIT Numerical Mathematics》2007,47(4):793-808
Linear algebra of factoring a matrix into the product of two matrices with special properties is developed. This is accomplished
in terms of the so-called inverse of a matrix subspace which yields an extended notion for the invertibility of a matrix.
The product of two matrix subspaces gives rise to a natural generalization of the concept of matrix subspace. Extensions of
these ideas are outlined. Several examples on factoring are presented.
AMS subject classification (2000) 15A23, 65F30 相似文献
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In this paper properties of cell matrices are studied. A determinant of such a matrix is given in a closed form. In the proof a general method for determining a determinant of a symbolic matrix with polynomial entries, based on multivariate polynomial Lagrange interpolation, is outlined. It is shown that a cell matrix of size n>1 has exactly one positive eigenvalue. Using this result it is proven that cell matrices are (Circum-)Euclidean Distance Matrices ((C)EDM), and their generalization, k-cell matrices, are CEDM under certain natural restrictions. A characterization of k-cell matrices is outlined. 相似文献
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Yun Fan 《Linear and Multilinear Algebra》2018,66(10):2119-2137
Double circulant matrices are introduced and studied. By a matrix-theoretic method, the rank r of a double circulant matrix is computed, and it is shown that any consecutive r rows of the double circulant matrix are linearly independent. As a generalization, multiple circulant matrices are also introduced. Two questions on square double circulant matrices are posed. 相似文献
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Raf Vandebril Marc Van Barel Nicola Mastronardi 《Numerical Linear Algebra with Applications》2005,12(7):625-658
The QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n × n matrix, this algorithm requires O(n3) operations per iteration step. To reduce this complexity for a symmetric matrix to O(n), the original matrix is first reduced to tridiagonal form using orthogonal similarity transformations. In the report (Report TW360, May 2003) a reduction from a symmetric matrix into a similar semiseparable one is described. In this paper a QR algorithm to compute the eigenvalues of semiseparable matrices is designed where each iteration step requires O(n) operations. Hence, combined with the reduction to semiseparable form, the eigenvalues of symmetric matrices can be computed via intermediate semiseparable matrices, instead of tridiagonal ones. The eigenvectors of the intermediate semiseparable matrix will be computed by applying inverse iteration to this matrix. This will be achieved by using an O(n) system solver, for semiseparable matrices. A combination of the previous steps leads to an algorithm for computing the eigenvalue decompositions of semiseparable matrices. Combined with the reduction of a symmetric matrix towards semiseparable form, this algorithm can also be used to calculate the eigenvalue decomposition of symmetric matrices. The presented algorithm has the same order of complexity as the tridiagonal approach, but has larger lower order terms. Numerical experiments illustrate the complexity and the numerical accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
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Hiroshi Hashimoto 《Fuzzy Sets and Systems》1984,12(2):155-168
A matrix operation is examined for fuzzy matrices and interesting properties of fuzzy matrices are obtained using the operation. Particularly some properties concerning subinverses and regularity of fuzzy matrices are given and the largest subinverse is shown by the properties. The properties are closely related to inverses of fuzzy matrices and fuzzy equations. Moreover fuzzy preorders are examined using the matrix operation and basic properties are obtained. The results are considered to be useful for the theory of fuzzy matrices. 相似文献