关于矩阵秩的一个不等式

对任意矩阵 M,用 r( M)表示 M的秩。熟知 ,矩阵的秩是矩阵的一个重要不变量 ,对矩阵的加法和乘法 ,我们有下面两个基本的不等式。(一 )设 A、B是两个 m× n矩阵 ,则r( A +B)≤ r( A) +r( B) ( 1 )　　 (二 )设 A、B分别是两个 m× n、n× l矩阵 ,则r( A) +r( B) -n≤ r( AB)≤ min{ r( A) ,r( B) }它通常被称为 Sylvester不等式。对这两个不等式 ,有不同的证明和理解 ,见 [1、2 ]。在本文里 ,我们要结合矩阵的满秩分解 ,用不等式 (二 )来研究不等式 (一 ) ,从中给出 r( A+B)≤ r( A) +r( B)的一个推广形式。本文所需的矩阵知识是基…     

一个猜想不等式的加细与推广

文 [1 ]提出如下猜想　设 x1,x2 ,… ,xn ∈ R+ ,x1+ x2 +… + xn =1 ,n≥ 3,n∈ N,则　 ∏ni=1( 1xi- xi)≥ ( n - 1n) n. ( 1 )戴承鸿、刘兵华在文 [2 ]中证明了上述猜想不等式成立 .本文给出该不等式的一个加细及推广形式 .定理　设 x1+ x2 +… + xn=k,n≥ 3,n∈ N;若 k≤ 1 ,x1,x2 ,… ,xn ∈ R+ ,则　 ∏ni=1( 1xi- xi)≥ ( nk - kn) n ( ∏ni=1nxik) 1n-13≥ ( nk - kn) n ( 2 )若 k≥ n - 1 ,x1,x2 ,… ,xn ∈ ( 0 ,1 ) ,则∏ni=1( 1xi- xi)≤ ( nk - kn) n .　　 ( ∏ni=1n - nxin - k) 13 -1n ≤ ( nk - kn) n. ( 3)为证定理 ,先…     

Erdos-Ko-Rado Theorems of Labeled Sets

For k = (k1, ··· , kn) ∈ Nn, 1 ≤ k1 ≤···≤ kn, let Lkr be the family of labeled r-sets on k given by Lkr := {{(a1, la1), ··· , (ar, lar)} : {a1, ··· , ar} ■[n],lai ∈ [kai],i = 1, ··· , r}. A family A of labeled r-sets is intersecting if any two sets in A intersect. In this paper we give the sizes and structures of intersecting families of labeled r-sets.     

基于受限等距性质的矩阵低秩稀疏逼近误差界

正1引言假设D∈R~(m×n)为实际观测到的高维数据矩阵,则从高维空间中估计一低维子空间的问题,称为矩阵低秩逼近,即估计一低秩矩阵A,使得D与A∈R~(m×n)之间的误差E=D-A最小化,该问题表示如下min‖E‖~2_F=‖D-A‖~2_F s.t.rank(A)≤r,其中r《min(m,n).求解矩阵低秩逼近问题最著名的方法是主成分分析法(Principal components analysis,PCA)[8,14,15],PCA在误差||E||_F较小的情况下,利用奇异值分解     

方阵秩与零特征值代数重数相关性探讨

利用方阵的特征方程和Jordan标准型,可得出方阵的秩R、零特征值代数重数r、几何重数s之间存在的两个关系式．即n-r≤R〈n和R=n-s．而这两个关系式又可用于简化方阵特征值和秩的求解．揭示矩阵零特征值代数重数与矩阵秩之间的内在必然联系．     

矩阵特征值的几个扰动定理

1 引言 设A∈C~(n×m),B∈C~(m×m)(m≤n),它们的特征值分别为{λ_k}_(k=1)~n和{μ_k}_(k=1)~m.令 R=AQ-QB (1)这里Q∈C~(n×m)为列满秩矩阵.Kahan研究了矩阵A在C~(n×m)上的Rayleigh商的性质,证明了下列定理:设A为Hermite矩阵,Q为列正交矩阵,即Q~HQ=I,而B=Q~HAQ,则存在 1,2,… ,n的某个排列π,使得 {sum from j=1 to m │μ_j-λ_(π(j))│~2}~(1/2)≤2~(1/2)‖R‖_F (2)其中R如(1)所示,‖·‖_F为矩阵的Frobenius范数.刘新国在[2]中将此定理推广到B为可对角化矩阵的情形,并且还建立了较为一般的扰动定理:设A为正规矩阵,B为可对角化矩阵;存在非奇异矩阵G,使得G~(-1)BG为对角阵,则存在1,2,…,n的某个排列π,使得 │μ_j-λ_(π(j))│≤2(2~(1/2))nK(G)_(σ_m~(-1))‖R‖_F,j=1,2,…,m. (3)     

关于矩阵方程求解的另一点注记

对任意给定的矩阵A∈Pm×n,B∈Pm×s(s≤n),探讨了矩阵方程AX=B有列满秩解,同时BY=A有行满秩解的充分必要条件,并且给出了基于矩阵的等价、齐次方程组的同解、向量组的等价及线性空间语言的推广.     

Saturation theorems for generalized Bernstein polynomials

In the present paper, we give the explicit formula of the principal part of n ∑ k=0 ([k]q -[n]qx)sxk n-k-1 ∏ m=0 (1-qmx) with respect to [n]q for any integer s and q ∈ (0,1]. And, using the expressions, we obtain saturation theorems for Bn(f,qn;x) approximating to f(x) ∈ C[0,1], 0 < qn ≤ 1, qn → 1.     

实对称矩阵在相合分解下的Moore-Penrose型广义逆的通式

1引言与符号说明对m×n矩阵A,下列矩阵方程：(1)AXA=A,(2)XAX=x,(3)(AX)~T=AX,(4)(XA)~T=XA称为Penrose方程．如果X满足上述方程(i)(j),…(k),则称X为(ij…k)逆,其全体记为A(ij…k)．(1234)逆常记为A~ ．所有这种矩阵叫广义逆(矩阵)或Moore- Penrose型逆(矩阵)．广义逆矩阵在许多数学领域有广泛应用．它在解矩阵方程中的作用     

布尔矩阵行空间基数的存在点

设Bn表示所有的n阶布尔矩阵的集合，R(A)表示A∈Bn的行空间．|R(A)|表示R（A）的基数。设m，n,k为正整数，本证明了当n≥9,[n 5/2]≤k≤n-3时,对任意的m、2^k≤m≤2^k 2^n-k 2 2^n-k 1 … 2^3，存在A∈B．使得|R（A）|=m．     

图中具有正交（g,f）因子分解的子图

设G是一个 (mg +k ,mf -k) -图 (1≤k 相似文献   

Twisted factorization of a banded matrix

The twisted factorization of a tridiagonal matrix T plays an important role in inverse iteration as featured in the MRRR algorithm. The twisted structure simplifies the computation of the eigenvector approximation and can also improve the accuracy. A tridiagonal twisted factorization is given by T=M _k Δ _k N _k where Δ _k is diagonal, M _k ,N _k have unit diagonals, and the k-th column of M _k and the k-th row of N _k correspond to the k-th column and row of the identity, that is . This paper gives a constructive proof for the existence of the twisted factorizations of a general banded matrix A. We show that for a given twist index k, there actually are two such factorizations. We also investigate the implications on inverse iteration and discuss the role of pivoting.      

与任意图正交的[0,k_i]_1~m-因子分解

设G是一个图 ,k1,… ,km 是正整数·　若图G的边能分解成m个边不交的 [0 ,k1]_因子F1,… ,[0 ,km]_因子Fm,则称 F =F1,… ,Fm 是G的一个 [0 ,ki]m1_因子分解·　如果H是G的一个有m条边的子图且对任意的 1≤i≤m有｜E(H) ∩E(Fi) ｜=1,则称 F与H正交·　证明了若G是一个 [0 ,k1 … km-m 1]_图 ,H是G的一个有m条边的子图 ,则图G有一个 [0 ,ki]m1_因子分解与H正交     

Wiener–Hopf factorization for a group of exponentials of nilpotent matrices

A complete study of the generalized factorization for a group of 2×2 matrix functions of the form G=I+γN, where , I denotes the 2×2 identity matrix and N represents a rational nilpotent matrix function, is presented. A closely related class involving the same matrix N is also studied. The canonical and non-canonical factorizations are considered and explicit formulas are obtained for the partial indices and the factors in such factorizations. It is shown in particular that only one of the columns in the factors needs to be determined, as a solution to a homogeneous linear Riemann–Hilbert problem, the other column being expressed in terms of the first. Necessary and sufficient conditions for existence of a canonical factorization within the same class are established, as well as explicit formulas for the factors in this case.     

Quadratic optimal control of stable well-posed linear systems

We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.

     

Cube factorizations of complete graphs

A cube factorization of the complete graph on n vertices, K_n, is a 3‐factorization of K_n in which the components of each factor are cubes. We show that there exists a cube factorization of K_n if and only if n ≡ 16 (mod 24), thus providing a new family of uniform 3‐factorizations as well as a partial solution to an open problem posed by Kotzig in 1979. © 2004 Wiley Periodicals, Inc.     

Modern factorization methods

In this expository paper the progress in factorization of large integers since the introduction of computers is reported. Thanks to theoretical advances and refinements, as well as to more powerful computers, the practical limit of integers possible to factor has been raised considerably during the past 20 years. The present practical limit is around 10⁷⁵ if supercomputers are used and if much computer time is available.     

A note on 3‐factorizations of K10*

In this note, we enumerate all nonisomorphic 3‐factorizations of K₁₀. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 379–383, 2001     

Constructively Factoring Linear Partial Differential Operators in Two Variables

We study conditions under which a partial differential operator of arbitrary order n in two variables or an ordinary linear differential operator admits a factorization with a first-order factor on the left.The process of factoring consists of recursively solving systems of linear equations subject to certain differential compatibility conditions.In the general case of partial differential operators, it is not necessary to solve a differential equation. In special degenerate cases, such as an ordinary differential operator, the problem eventually reduces to solving some Riccati equation(s). We give the factorization conditions explicitly for the second and third orders and in outline form for higher orders. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 165–180, November, 2005.