Errors in matrices may also be measured with norms . Here is some related notation we will use in our error bounds. The condition number of a matrix A is defined as , where A is square and invertible, and p is or one of the other possibilities in Table 4.2 . The reciprocal of the condition number is used instead of the condition number itself in order to avoid the possibility of overflow when the condition number is very large. Now we consider errors in subspaces .
Subspaces are the outputs of routines that compute eigenvectors and invariant subspaces of matrices.
We need a careful definition of error in these cases for the following reason. In other words, eigenvectors are not unique. This means we cannot measure the difference between two supposed eigenvectors and x by computing , because this may be large while is small or even zero for some . So in order to define error in a useful way, we need to instead consider the set of all scalar multiples of x . The set is called the subspace spanned by x , and is uniquely determined by any nonzero member of .