Linear Algebra with Applications, 10th Edition

6.5 The Singular Value Decomposition

In many applications, it is necessary either to determine the rank of a matrix or to determine whether the matrix is deficient in rank. Theoretically, we can use Gaussian elimination to reduce the matrix to row echelon form and then count the number of nonzero rows. However, this approach is not practical in finite-precision arithmetic. If A is rank deficient and U is the computed echelon form, then, because of rounding errors in the elimination process, it is unlikely that U will have the proper number of nonzero rows. In practice, the coefficient matrix A usually involves some error. This may be due to errors in the data or to the finite number system. Thus, it is generally more practical to ask whether ...