11.3 The Norm and the Condition Number

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	Java Number Cruncher: The Java Programmer's Guide to Numerical Computing By Ronald  Mak
	Table of Contents
	Chapter  11.   Matrix Inversion, Determinants, and Condition Numbers

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11.3 The Norm and the Condition Number

How do we measure how well-conditioned or ill-conditioned is a system of linear equations? In Chapter 10, we said that the solution to an ill-conditioned system is very sensitive to changes (perhaps due to roundoff errors) in the values of the coefficients.

The condition number of the system's coefficient matrix A is defined to be

where A is the norm of the matrix A, and A ^-1 is the norm of its inverse. The Euclidean norm of any square matrix A is the square root of the sum of the squares of all its elements:

A well-conditioned system has a "small" condition number, and an ill-conditioned matrix has a "large" condition number. This is a somewhat fuzzy measure?athere are no hard and fast rules about how small a condition number must be before we can say for sure that a system is well-conditioned, and how large a condition number must be before we can say for sure that a system is ill-conditioned. We can use the number to compare the condition of two similarly sized systems. We'll see some examples of this in the following paragraphs.

For example, the norm of the 3x3 identity matrix I ₃ is . Since I ₃ ^{- 1} = I ₃ ,

which is considered small, and, indeed, an identity matrix is extremely well-conditioned.

Note that the condition number of a matrix is somewhat expensive to compute, since it requires computing the inverse of the matrix.

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