共查询到19条相似文献,搜索用时 123 毫秒
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定义在正整数集合上的复值函数称为算术函数.本文讨论算术函数的两种多元扩张及其对GCD函数矩阵与LCM函数矩阵的应用. 相似文献
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将对角占优矩阵的性质与矩阵的直积结合起来,给出了两矩阵的直积是对角占优矩阵的一些充分和必要条件,推广了近期的一些结果.最后用相应的数值例子说明了所得结果的有效性. 相似文献
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关于随机矩阵Kronecker积的谱半径的不等式 总被引:2,自引:0,他引:2
研究了随机矩阵的Kronecker积的数学期望的性质,得到了随机矩阵的Kronecker积的谱半径的几个不等式. 相似文献
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设S={x1,x2,...,xn}是由n个不同的正整数组成的集合,并设a为正整数.如果一个n阶矩阵的第i行j列元素是S中元素xi和xj的最大公因子的a次幂(xi,xj)a,则称该矩阵为定义在S上的a次幂最大公因子(GCD)矩阵,用(Sa)表示;类似定义a次幂LCM矩阵[Sa].如果存在{1,2,...,n}上的一个置换σ使得xσ(1)|xσ(2)|···|xσ(n),则称S为一个因子链.如果存在正整数k,使得S=S1∪S2∪···∪Sk,其中每一个Si(1ik)均为一个因子链,并且对所有的1i=jk,Si中的每个元素与Sj中的每个元素互素,则称S由有限个互素因子链构成.本文中,设S由有限个互素的因子链构成,并且1∈S.我们首先给出幂GCD矩阵与幂LCM矩阵的行列式的公式,然后证明:如果a|b,则det(Sa)|det(Sb),det[Sa]|det[Sb],det(Sa)|det[Sb].最后我们指出:如果构成S的有限个因子链不互素,则此结论一般不成立. 相似文献
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设S={x1,x2,…xn}是一个由非零整数且|xi|≠|xj≠k.1≤i,j≤n)组合的集合,我们先定义了集S上的广义GCD(GGCD)矩阵和广义LCM(GLCM)矩阵,然后计算了定义在广义gcd-closed集上的GGCD矩阵和CLCM矩阵的逆矩阵。 相似文献
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Pentti Haukkanen 《Linear and Multilinear Algebra》2013,61(3):301-309
Considering lower closed sets as closed sets on a preposet (P, ≤), we obtain an Alexandroff topology on P. Then order preserving functions are continuous functions. In this article we investigate order preserving properties (and thus continuity properties) of integer-valued arithmetical functions under the usual divisibility relation of integers and power GCD matrices under the divisibility relation of integer matrices. 相似文献
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Pauliina Ilmonen 《Linear and Multilinear Algebra》2013,61(5):557-564
In 1861, Henry John Stephen Smith [H.J.S. Smith, On systems of linear indeterminate equations and congruences, Philos. Trans. Royal Soc. London. 151 (1861), pp. 293–326] published famous results concerning solving systems of linear equations. The research on Smith normal form and its applications started and continues. In 1876, Smith [H.J.S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875/76), pp. 208–212] calculated the determinant of the n?×?n matrix ((i,?j)), having the greatest common divisor (GCD) of i and j as its ij entry. Since that, many results concerning the determinants and related topics of GCD matrices, LCM matrices, meet matrices and join matrices have been published in the literature. In this article these two important research branches developed by Smith, in 1861 and in 1876, meet for the first time. The main purpose of this article is to determine the Smith normal form of the Smith matrix ((i,?j)). We do this: we determine the Smith normal form of GCD matrices defined on factor closed sets. 相似文献
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Some analogues of smith's determinant 总被引:1,自引:0,他引:1
We calculate the determinants of the greatest common divisor (GCD) and the least common multiple (LCM) matrices associated with an arithmetical function on gcd-closed and lcm-closed sets. We also consider some unitary analogues of these determinants. 相似文献
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广义酉矩阵与广义Hermite矩阵的张量积与诱导矩阵 总被引:2,自引:0,他引:2
本文研究了有限个广义酉矩阵与广义(反)Hermite矩阵的张量积和诱导矩阵.利用矩阵的张量积和诱导矩阵的性质,得到了它的张量积和诱导矩阵仍为广义酉矩阵与广义(反)Hermite矩阵. 相似文献
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Raphaël M. Jungers 《Linear algebra and its applications》2008,428(10):2283-2295
We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This “finiteness conjecture” is now known to be false but no explicit counterexample is available and in particular it is unclear if a counterexample is possible whose matrices have rational or binary entries. In this paper, we prove that all finite sets of nonnegative rational matrices have the finiteness property if and only if pairs of binary matrices do and we state a similar result when negative entries are allowed. We also show that all pairs of 2×2 binary matrices have the finiteness property. These results have direct implications for the stability problem for sets of matrices. Stability is algorithmically decidable for sets of matrices that have the finiteness property and so it follows from our results that if all pairs of binary matrices have the finiteness property then stability is decidable for nonnegative rational matrices. This would be in sharp contrast with the fact that the related problem of boundedness is known to be undecidable for sets of nonnegative rational matrices. 相似文献
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In this article, we study some algebraic and geometrical properties of polynomial numerical hulls of matrix polynomials and joint polynomial numerical hulls of a finite family of matrices (possibly the coefficients of a matrix polynomial). Also, we study polynomial numerical hulls of basic A-factor block circulant matrices. These are block companion matrices of particular simple monic matrix polynomials. By studying the polynomial numerical hulls of the Kronecker product of two matrices, we characterize the polynomial numerical hulls of unitary basic A-factor block circulant matrices. 相似文献
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从y-数值半径的定义出发,利用矩阵张量积与诱导矩阵的性质,研究了它们的y-数值半径,得到了矩阵张量积与诱导矩阵y-数值半径的几个不等式. 相似文献