p,j], which turns out to be the

augmented matrix [[V.

This system has solutions according to the KroneckerCapelli theorem (rank of this system

augmented matrix and rank of this system basic matrix equal to 7).

Step 1 : Starting with the augmented matrix (A | B), reduce it to a form with an identity matrix of desired order, by elementary row operations only.

We thus start with the augmented matrix (A | B) and proceed as follows, using the indicated row operations:

For example, to compute a few of the smallest singular values, MATLAB's routine svds applies ARPACK [30, 39] to the augmented matrix M (1.

It appears that the augmented matrix formulation (1.

The program j dsvd [21, 22] implements a Jacobi-Davidson method on the augmented matrix formulation.

T] decomposition of the symmetric augmented matrix is computed using preprocessing that will help to minimize the fill-in during the factorization combined with one by one and two by two numerical pivot strategies [18, 17].

In Section 2, we will briefly summarize the approximation process and describe the basic properties of the linear system and augmented matrix.

where the elements of the matrices M and A of the augmented matrix of the left-hand side of (2.

1] after the row and column permutations, the augmented matrix in (2.

A portion of the singular values and vectors of a large sparse matrix A is computed in [12] by using a Jacobi-Davidson type method which makes use of the block structure of the

augmented matrix