Euclidean norm and condition number of a square matrix

6 Euclidean norm and condition number of a square matrix 

In Section 1.7 we emphasized the importance of the condition number cond(A), but did not tell you how to find it. Now we try to close this gap, first 

This problem is easy to solve by writing Ax in the basis of the 

Since ATA is symmetric and positive semidefinite, x is real and nonnegative. If the matrix is nonsingular (and hence positive definite) then 

We note that the values X /2 are called the singular values of the matrix A and they can be determined directly from A, without forming ATA. The corresponding numerical method called singular value decomposition is relatively complex but somewhat more accurate then the procedure described here, for details see (ref. 11). 

---