袁錦昀教授是杰出的旅居巴西華人1957年出生于江蘇興化唐劉鎮(zhèn)，1977年考入南京工學(xué)院，巴西巴拉那聯(lián)邦大學(xué)數(shù)學(xué)系終身教授、工業(yè)數(shù)學(xué)研究所所長，巴西計算和應(yīng)用數(shù)學(xué)學(xué)會副會長，巴西數(shù)學(xué)會巴拉那州分會會長，巴西科技部基金委數(shù)學(xué)終審組應(yīng)用數(shù)學(xué)和計算數(shù)學(xué)負(fù)責(zé)人，巴西巴拉那基金委數(shù)學(xué)終身組成員。 《實用迭代分析(英文版)(精)》是由其創(chuàng)作的英文版實用迭代分析專著。
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Chapter 1
Introduction
In this chapter, we first give an overview of relevant concepts and some basic results of matrix linear algebra. Materials contained here will be used throughout the book.
1.1 Background in linear algebra
1.1.1 Basic symbols, notations, and definitions
Let R be the set of real numbers; C,the set of complex numbers; and i 三 \/^T. The symbol Rn denotes the set of real n-vectors and Cn, the set of complex n-vectors, a, /3, 7,etc., denote real numbers or constants. Vectors are almost always column vectors. We use Rmxn to denote the linear vector space of all m-by-n real matrices and Cmxn, the linear vector space of all m-by-n complex matrices. The symbol dimp) denotes the dimension of a linear vector space S.
The upper case letters A, B, C, A, A, etc., denote matrices and the lower case letters x, y, z, etc., denote vectors.
Let (A)ij = ctij denote the (i, j)th entry in a matrix A = (aij). For any n-by-n matrix, the indices j go through 1 to n usually but sometimes go through 0 to n — 1 for convenience. Let AT be the transpose of A; A*, the conjugate transpose of
A\ rank(yl), the rank of A\ and tr(A)三the trace of A. An n-by-n diagonal
matrix is denoted by
We use the notation In for the n-by-n identity matrix. When there is no ambiguity, we shall write it as I. The symbol ej denotes the jth unit vector, i.e., the jth column vector of I.
A matrix A G Rnxn is symmetric if AT = A, and skew-symmetric if AT = —A. A symmetric matrix A is positive definite (semidefinite) if xTAx > 00) for any
nonzero vector x G Rn. A matrix A G Cnxn is Hermitian if A* = A. A Hermitian matrix A is positive definite (semidefinite) if x*Ax ≥ 0( 0) for any nonzero vector
x e Cn.
A number A e C is an eigenvalue of A G Cnxn if there exists a nonzero vector x G Cn such that Ax = Xx, where x is called the eigenvector of A associated with A. It is well-known that the eigenvalues of all Hermitian matrices are real. Let Amin (A) and Amax(A) denote the smallest and largest eigenvalues of a Hermitian matrix A respectively. We use p(A) = max |Ai(A)| to denote the spectral radius of A where Ai(A) goes through the spectrum of A. Recall that the spectrum of A is the set of all the eigenvalues of A.
We use to denote a norm of vector or matrix. The symbols||oo denote the p-novm with p = 1,2, oo, respectively. Also we use ?a(A), which is defined by Ka(A) = ||A||a||A_1||a to denote the condition number of the matrix A. In general, we consider every norm at the definition when a is omitted. But most used norm is 2-norm.
We use and 1Z(A) to represent the null space and Image space (or Range)
of given matrix A respectively where = {x G Rn : Ax = 0} and 1^(A) = {y G
Rm : y = Ax for some x G Rn} and A G Rmxn.
For matrix iterative analysis, we need some tools, such as vector norms, matrix norms and their extensions, and spectral radii.
1.1.2 Vector norm
In fact, a norm is an extension of length of vector in R2 or absolute value in R. It is well-known that Vx G R, \x\ = satisfies the following properties:
We generalize three properties above to vector space Rn as follows.
Definition 1.1.1 /i : Rn —j- R is a vector norm on Rn if
Example 1.1.1 There are three common norms on Rn defined by
There axe some important elementary consequences from Definition 1.1.1 of the vector norm.
Proposition 1.1.1
Proof
Then,
By interchanging x and y, we can obtain
The result of (1.1.1) follows from (1.1.3) and (1.1.4) together. We can prove (1.1.2) if y is replaced by —y in (1.1.1).
The 2-norm is the natural generalization of the Euclidean length of vector on R2 or R3 and called the Euclidean norm. The oo-norm also sometimes called the maximum norm or the Chebyshev norm. In fact, they are special cases of p-norm defined as ,
Sometimes, usual norm is not enough for our research. We have to construct a new norm. One useful technique to construct new norms from some well-known norm is given in the following theorem.
Theorem 1.1.2 Let v be a norm on Rm and A E Rmxn have linearly inde?pendent columns. Then /i(x) = u(Ax) : Rn is a norm on Rn.
The proof is easy, just to check properties of the norm in Definition 1.1.1. Leave it to reader. This technique can work for matrix norm in the next subsection.
Corollary 1.1.3 Let A G RnXn be symmetric and positive definite. Then, /i(x) = VxTAx is a norm on Rn? denoted ||尤||^4,and called weighted norm (with A). We have to know if the sequence generated by iterative methods converges to the solution when we study iterative methods. For this purpose, we shall give some concepts about limit of sequence in vector spaces.
Definition 1.1.2 Let {x(fc)} be a sequence of n-vectors,and x G Rn. Then, x is a limit of the sequence {x(fc)} (written x = limfc_,00 x^) if
where Xi(i = 1,2, … ,n) are components of x.
By the definition,
Furthermore, it follows from equivalence of vector norms that x = lim a;⑷ lim "(x — a:⑷)=0,
where " is a norm on Rn.
1.1.3 Matrix norm
Definition 1.1.3 : Rmxn — R is a matrix norm on Rmxn if
Example 1.1.2 Let A
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