Linear Algebra with Applications, 10th Edition

7.4 Matrix Norms and Condition Numbers

In this section, we are concerned with the accuracy of computed solutions of linear systems. How accurate can we expect the computed solutions to be, and how can we test their accuracy? The answer to these questions depends largely on how sensitive the coefficient matrix of the system is to small changes. The sensitivity of the matrix can be measured in terms of its condition number. The condition number of a nonsingular matrix is defined in terms of its norm and the norm of its inverse. Before discussing condition numbers, it is necessary to establish some important results regarding the standard types of matrix norms.

Matrix Norms

Just as vector norms are used to measure the size of vectors, matrix norms ...