Chapter 7

Vector and Matrix Norms

Abstract

This chapter begins the study of numerical linear algebra. The length, or norm, of a vector is defined, and the 2-, 1-, and infinity vector norms are defined. There are inequalities that bound each norm in terms of the others. The chapter develops properties of the 2-norm, since it is most frequently used in applications. In particular, the Cauchy-Schwarz inequality and the Pythagorean theorem are satisfied by the 2-norm. The 2-norm is orthogonally invariant and thus is very important in computer graphics. If a set of k vectors in n-dimensional space are orthogonal, they form a basis for a k-dimensional subspace. Multiplication by an orthogonal matrix effects a change of coordinates. Spherical coordinates ...

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