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Ans 3 - v1 - EE448/528 Version 1.0 John Stensby Chapter 4...

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EE448/528 Version 1.0 John Stensby CH4.DOC Page 4-1 Chapter 4: Matrix Norms The analysis of matrix-based algorithms often requires use of matrix norms. These algorithms need a way to quantify the "size" of a matrix or the "distance" between two matrices. For example, suppose an algorithm only works well with full-rank, n × n matrices, and it produces inaccurate results when supplied with a nearly rank deficit matrix. Obviously, the concept of e - rank (also known as numerical rank ), defined by rank A rank B A B ( , ) min ( ) ε ε = - (4-1) is of interest here. All matrices B that are "within" ε of A are examined when computing the e - rank of A. We define a matrix norm in terms of a given vector norm; in our work, we use only the p- vector norm, denoted as r X p . Let A be an m × n matrix, and define A AX X p X p p = sup r r r 0 , (4-2) where "sup" stands for supremum, also known as least upper bound. Note that we use the same p notation for both vector and matrix norms. However, the meaning should be clear from context. Since the matrix norm is defined in terms of the vector norm, we say that the matrix norm is subordinate to the vector norm. Also, we say that the matrix norm is induced by the vector norm. Now, since r X p is a scalar, we have A AX X X X p X p p X p p = = sup sup / r r r r r r 0 0 A . (4-3)
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EE448/528 Version 1.0 John Stensby CH4.DOC Page 4-2 In (4-3), note that r r X X p / has unit length; for this reason, we can express the norm of A in terms of a supremum over all vectors of unit length, A X p X p p = = sup r r 1 A . (4-4) That is, A p is the supremum of AX p r on the unit ball r X p = 1. Careful consideration of (4-2) reveals that AX A X p p p r r (4-5) for all X r . However, AX p r is a continuous function of X r , and the unit ball r X p = 1 is closed and bounded (real analysis books, it is said to be compact ). Now, on a closed and bounded set, a continuous function always achieves its maximum and minimum values. Hence, in the definition of the matrix norm, we can replace the "sup" with "max" and write A AX X X p X p p X p p = = = A max max r r r r r 0 1 . (4-6) When computing the norm of A, the definition is used as a starting point. The process has two steps. 1) Find a "candidate" for the norm, call it K for now, that satisfies AX X p p r r K for all X r . 2) Find at least one nonzero X r 0 for which AX X p p r r 0 0 = K Then, you have your norm: set A p = K. MatLab's Matrix Norm Functions From an application standpoint, the 1-norm, 2-norm and the -norm are amoung the most
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EE448/528 Version 1.0 John Stensby CH4.DOC Page 4-3 important; MatLab computes these matrix norms. In MatLab, the 1-norm, 2-norm and -norm are invoked by the statements norm(A,1) , norm(A,2) , and norm(A,inf) , respectively. The 2-norm is the default in MatLab. The statement norm(A) is interpreted as norm(A,2) by MatLab. Since the 2-norm used in the majority of applications, we will adopt it as our default. In what follows, an "un-designated" norm A is to be intrepreted as the 2-norm A 2 .
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