積分∫F(x,y,y′,…,y~((n)))dx的等價問题

<正> 命g為x,y平面上由所有保切變換所構成的羣。在本文內我們將定義一類廣義空間使這空間與積分∫F(x,y,y′,…y~((n)))dx對於羣g而言有不變的聯繫。所謂一空間對於羣g而言與積分∫Fdx有不變的聯繫,意義是:如我們施用羣g     

多個複變數函數論Ⅱ 超球雙曲空間中的一完整正交函數系

<正> §1.引言 在上一篇文章裹,我曾經具體地算出矩陣的雙曲空間中的完整正交函数系,在該文中引用了方陣羣的表示法的理論.在這一篇文章裹,我們將定出超球雙曲空間中的完整正交系.所用的方法和上篇稍有不同,我們除掉用一些正交羣的表示羣以外,還用了不變量論中的結果及若干與球調和(spherical harmonic)     

原始遞歸函數的定義

<正> 符號說明 本文中除以α表示常數外均以小寫拉丁字母表示自變數,以大寫字母表示函數關係,而一元函數A(x)常省寫為A_x. 在本文中只討論數論函數,自口自變數与函數所取之值均限於正整數或0(常     

廣義Stieltjes-Post反演公式及漸近積分

<正> 在作者之一的文章[1]中,曾定義過一種含有参數的正規變換函數類。對於以這類中的函數為核所構成的積分變換,即存在有一種廣義的Stieltjes-Post-Widder反演公式。在本文的第一節中,我們將對正規變換函數定義中的第二條件予以减弱,也就是把核函數的範圍加以放寬,而仍保持廣義反演公式的有效.在本文的第二節中,主要是改善先前一篇短文[2]中的結果,我們將在較廣泛的條件下,重新建立某一漸近積分定理.     

經驗分佈之比的極限分佈

<正> §1.引言 令x為一隨機變數,其分佈函數為F(x).對於x作n次相互獨立的试驗,便得n個結果x_1,x_2,…,x_n.我們也可以把x_1,x_2,…,x_n看作是遵循同一個分佈函數F(x)的相互獨立隨機變數.現在把x_1,x_2,…,x_n依其值由小到大的次序排列,我們得到     

初等变换的“記录矩陣法”

設A是一个n阶非异方陣,我們可用下法来求A的逆方陣A~(-1),即在A的右方列一个n阶单位方陣E,得到一个n×2n矩陣,对这个矩陣作初等行变換使前n列变为E則后n阶此时即組成A~(-1)。这个方法在許多綫性代数教科书中均可找到。我們不妨称这个方法为“記录矩陣法”。这个方法甚为簡捷。本文中我們来研究这个方法在向量問題及綫性方程組中的一些应用。可以看出,在这些問題之应用中本法仍不失为一个簡捷的計算法。     

論齊性空間內廣義球函數的正交性及完整性

<正> 1.引言.設為一空間,為給定的一個羣.羣中每一元素σ可以看做是空間裹一個變换:對於中每一點P,對應着一個且唯一的點,記做σP.σ就簡稱做把點P移到點σP的一個變換,它也可表示如下:     

辛空間微分幾何引論

<正> 本文將分為兩個組成部分.第一部分討論普通的辛空間,就是以辛变換羣為基本羣的空間.特別地,我們將討論這空間內曲線與曲面的微分幾何.第二部分討論所謂“辛流空間”.     

無窮小

無窮小是以零為極限的變量,對於這種變量引用普遍的極限定義可得如下的深入的定義:若對於任何指定的正數ε,變量y在其變化過程中達到這樣一個時刻,從該時刻起y的絕對值恒保持小於ε,則變量y稱為無窮小,如n依次取所有的自然數1,2,3,…时,變量為無窮小,因為當n>100時y<0.1,當n>10000時y<0.01,一般當n>1/ε~2时y<ε。若變量y的極限為有盡數a,则極限式limy=a相當於關係式lim(y-a)=0,即相當於差(y-a)為無窮小,因此我們也可以反過來:把無窮小的定義—特殊場合     

給“數学教學”編者的信

在第30版■吉西略夫的代數教科書中的第57頁上,所叙述的雙曲線定義,能够把學生引入迷路,就是:“函數y=k/x的圖象稱為雙曲線。當k與x為正值時,雙曲線在第一象限,但當k為負而x為正時,它再第四象限,當變數x為負值時,即得雙曲線的另一枝,當k>0它在第三象限,但當k<0它在第二象限。”把參數與自變數的值在一句話中混淆起來,無論就科學的或是教學法的觀點來說,都是不允許的,這樣只能使學生糊塗。教本中的這個地方應該如下地叙述: “函數y=k/x的圖象稱為雙曲線。首先假定k為正,於是當x的值為正時,對應的y值也為正,而我們得到雙曲線的點在第一象限內,當x的值為負時,雙曲線的點在第三象限內。由於對於x=0的值,任何y的值都不能與之對應,所以在縱軸上沒有雙曲線的點;因此整個曲線分成兩枝,一枝在第一象限而另一枝在第三象限。     

Hadamard定理在四元数体上的推广

Let K be any skew-field with central field F. A matrix A=(a_ij)n×n over K is called centralized if the characteristic matrix λI-A can be reduced by some elementary transformations into the following diagonal form:such that are all manic polynomials over F. The determinant of a centralized matrix A=(a_ij)n×n may be defined by (1) asfollows: and then the famous theorem of Hadamard can be generalized as in the following: Theorem. If A=(a_ij)n×n is a non-singular centralized matrix over the skewfield of quaternions, thenand the equality sign holds if and only if the columns of A are all muturally orthogonal. This theorem may be proved by the following three lemmas: Lemma 1. If A is a centralized n-rowed square matrix of quaternions, then sois and Lemma 2. For any n-rowed square, matrix A of quaternions, the. following two matrices are always centralized:and Lemma 3. If A is a centralized matrix of quaternions, then we have     

THE SINE TRANSFORM OPERATOR IN THE BANACH SPACE OF SYMMETRIC MATRICES AND ITS APPLICATION IN IMAGE RESTORATION

In this paper, we study an operator s which maps every n-by-n symmetric matrix A, to a matrix s(A_n) that minimizes || B_n-A_n || F over the set of all matrices B_n, that can be diagonalized by the sine transform. The matrix s(A_n), called the optimal sine transform preconditioner, is defined for any n-by-n symmetric matrices A_n. The cost of constructing s(A_n) is the same as that of optimal circulant preconditioner c(A_n) which is defined in [8], The s(A_n) has been proved in [6] to be a good preconditioner in solving symmetric Toeplitz systems with the preconditioned conjugate gradient (PCG) method. In this paper, we discuss the algebraic and geometric properties of the operator s, and compute its operator norms in Banach spaces of symmetric matrices. Some numerical tests and an application in image restoration are also given.     

The Nearest Bisymmetric Solutions of Linear Matrix Equations

The necessary and sufficient conditions for the existence of and the expressions for the bisymmetric solutions of the matrix equations (Ⅰ)A1X1B1 A2X2B2 ^… AkXkBk=D,(Ⅱ)A1XB1 A2XB2 … AkXBk=D and (Ⅲ) (A1XB1,A2XB2,…,AkXBk)=(D1,D2,…,Dk) are derived by using Kronecker product and Moore-Penrose generalized inverse of matrices. In addition, in corresponding solution set of the matrix equations, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm is given. Numerical methods and numerical experiments of finding the neaxest solutions axe also provided.     

On a family of matrix equalities that involve multiple products of generalized inverses

Aequationes mathematicae - One of the fundamental matrix equalities that involve multiple products of matrices and their generalized inverses is given by $$A_1B_1^{-}A_2B_2^{-} \ldots...     

关于矩阵群逆的逆序律

得到了体上两个n阶方阵A,B的群逆A#,B#若存在,则其乘积的群逆(AB) #也存在,且(AB) #=B#A#成立的充分与必要条件是:存在n阶可逆矩阵P使得A =Pdiag(A1,A2 ,…,As) P- 1,B =Pdiag(B1,B2 ,…,Bs) P- 1且对于任意i(i=1 ,2 ,…,s)有Ai,Bi阶数相同,Ai,Bi为可逆矩阵或为0矩阵;又对i≠1有Ai Bi=0 .     

Miniversal deformations of matrices of bilinear forms

Arnold [V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26 (2) (1971) 29–43] constructed miniversal deformations of square complex matrices under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We construct miniversal deformations of matrices under congruence.     

矩阵对的相似标准形

设A,B,C,D都是n阶方阵,矩阵对(A,B)相似于矩阵对(C,D),如果存在n阶可逆矩阵P,使得P-1AP=C,P-1BP=D.本文借助Belitskii约化算法,提供一种在相似变化下化任一n阶矩阵对为标准形的有效方法,该方法可以看作Jordan标准形的推广.     

Quadratically normal and congruence-normal matrices

A matrix A ∈ C ^n×n is unitarily quasidiagonalizable if A can be brought by a unitary similarity transformation to a block diagonal form with 1 × 1 and 2 × 2 diagonal blocks. In particular, the square roots of normal matrices, i.e., the so-called quadratically normal matrices are unitarily quasidiagonalizable. A matrix A ∈ C ^n×n is congruence-normal if B = A[`(A)] B = A\overline A is a conventional normal matrix. We show that every congruence-normal matrix A can be brought by a unitary congruence transformation to a block diagonal form with 1 × 1 and 2 × 2 diagonal blocks. Our proof emphasizes andexploitsalikenessbetween theequations X <sup>2</sup> = B and X[`(X)] = B X\overline X = B for a normal matrix B. Bibliography: 13 titles.     

Characterizations of optimal scalings of matrices

A scaling of a non-negative, square matrixA ≠ 0 is a matrix of the formDAD ^?1, whereD is a non-negative, non-singular, diagonal, square matrix. For a non-negative, rectangular matrixB ≠ 0 we define a scaling to be a matrixCBE ^?1 whereC andE are non-negative, non-singular, diagonal, square matrices of the corresponding dimension. (For square matrices the latter definition allows more scalings.) A measure of the goodness of a scalingX is the maximal ratio of non-zero elements ofX. We characterize the minimal value of this measure over the set of all scalings of a given matrix. This is obtained in terms of cyclic products associated with a graph corresponding to the matrix. Our analysis is based on converting the scaling problem into a linear program. We then characterize the extreme points of the polytope which occurs in the linear program.     

Generating equations approach for quadratic matrix equations

We show how Van Loan's method for annulling the (2,1) block of skew‐Hamiltonian matrices by symplectic‐orthogonal similarity transformation generalizes to general matrices and provides a numerical algorithm for solving the general quadratic matrix equation: For skew‐Hamiltonian matrices we find their canonical form under a similarity transformation and find the class of all symplectic‐orthogonal similarity transformations for annulling the (2,1) block and simultaneously bringing the (1,1) block to Hessenberg form. We present a structure‐preserving algorithm for the solution of continuous‐time algebraic Riccati equation. Unlike other methods in the literature, the final transformed Hamiltonian matrix is not in Hamiltonian–Schur form. Three applications are presented: (a) for a special system of partial differential equations of second order for a single unknown function, we obtain the matrix of partial derivatives of second order of the unknown function by only algebraic operations and differentiation of functions; (b) for a similar transformation of a complex matrix into a symmetric (and three‐diagonal) one by applying only finite algebraic transformations; and (c) for finite‐step reduction of the eigenvalues–eigenvectors problem of a Hermitian matrix to the eigenvalues– eigenvectors problem of a real symmetric matrix of the same dimension. Copyright © 1999 John Wiley & Sons, Ltd.