Matrix norms

Each of the above vector norms induces a corresponding norm on matrices, through the definition 

It is possible to deduce from this definition the three properties (a),(b) and (c) above, and also a fourth property, (d), that AB 7i B.  

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Vector and Matrix Norms To carry out error analysis for approximate and iterative methods for the solutions of linear systems, one needs notions for vec tors in iT and for matrices that are analogous to the notion of length of a geometric vector. Let R denote the set of all vec tors with n components, x = x, . . . , x ). In dealing with matrices it is convenient to treat vectors in R as columns, and so x = (x, , xj however, we shall here write them simply as row vectors. 

An important tool in the study of matrices is provided by vector norms and matrix norms. A vector norm j... is any real valued function of the elements satisfying the following three conditions ... 

A square matrix of order n can be considered a vector in a space of n2 dimensions, and if the matrix function j... defines a vector norm in 2-space, it will be called a generalized matrix norm. Thus a generalized matrix norm satisfies... 

A matrix function v(...) will be called a mvltiplioative (strictly, submultiplicative) matrix norm, or simply a matrix norm, if it is a generalized matrix norm, and satisfies also... 

Given any generalized matrix norm. .., for sufficiently large, fixed a > 0,... 

A matrix norm is consistent (or compatible) with a given vector norm provided... 

Associated with every vector norm, there is a special consistent matrix norm defined as follows ... 

Since every matrix norm is consistent with some vector norm, the following important theorem follows immediately ... 

First, suppose p(B) < 1. Then there exists a matrix norm with v(B) < 1. Since the roots oil — B and of B can be paired so that... 

The euclidean norm of a matrix considered as a vector in m2-space is a matrix norm that is consistent with the euclidean vector norm. This is perhaps the matrix norm that occurs most frequently in the literature. But the euclidean norm of I is n112 > 1 when n > 1, hence it is not a sup. In fact,... 

Vector and Matrix Norms To carry out error analysis for... 

The matrix norm may be calculated by using one of the following formulas ... 

The definition above of a matrix norm is not directly evaluable in finite time. However, it is possible to determine the value of each of the norms from the elements of a matrix without working through all possible vectors. 

The norm of a matrix is said to be compatible with a given vector norm provided Eq. (A-29) is satisfied for any matrix A and any conformable vector X. The following procedure is employed for the construction of the matrix norm such that it is compatible with a given vector norm. First, observe that the lengths of all vectors in any ri-dimensional vector space span the set of all real numbers, and that the process of normalization of each vector throws the lengths of all... 

First it will be shown that for any real nonsingular matrix A of order n, there exists a particular vector Xx such that jXj = 1 and flAXj = A. The matrix norm so constructed is said to be subordinate to the given vector norm. Then it will be shown that the matrix norm so constructed satisfies the four conditions required of a matrix norm as well as the compatibility relationship. Only the matrix norm subordinate to the euclidean vector norm X , given by Eq. (A-26) is considered. Now consider the vector AX, whose vector norm as given by Eq. (A-26), is... 

The matrix norm A = y/T, where is the largest eigenvalue of H = A7A, which is commonly referred to as the Hilbert or spectral norm of matrix A. 

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