TABLE OF CONTENTS While it is true that all norms are equivalent theoretically, only a homely one like the ?-norm is truly useful numerically.
-J. H. WILKINSON, [9] Lecture at Stanford University (1984)
Matrix norms are defined in many different ways in the older literature, but the favorite was the Euclidean norm of the matrix considered as a vector in n 2-space.
Wedderburn (1934) calls this the absolute value of the matrix and traces the idea back to Peano in 1887.
-ALSTON S. HOUSEHOLDER,
The Theory of Matrices in Numerical Analysis (1964)
[9]Quoted in Fox [439, 1987].
Norms are an indispensable tool in numerical linear algebra. Their ability to compress the mn numbers in an m n matrix into a single scalar measure of size enables perturbation results and rounding error analyses to be expressed in a concise and easily interpreted form. In problems that are badly scaled, or contain a structure such as sparsity, it is often better to measure matrices and vectors componentwise. But norms remain a valuable instrument for the error analyst, and in this chapter we describe some of their most useful and interesting properties.
A vector norm is a function : 
satisfying the following conditions:
x ? 0 with equality iff x = 0.
? x = ? x for all 
.
x + y ? x...
