If the rank of B's is the same as that of the

augmented matrix (B's|y), B'sa' = y is a consistent equation, and there exist infinite solution a'=[B'.

and the

augmented matrix have the same rank; thus, at least one solution exists.

After many trials

augmented matrix of size M x N, given by equation (3), formed by combining identity matrix of size M x M, and zero matrix of size M x (N - M), was found to extract necessary information selectively since it has unity values in the leading diagonal.

The substitution point of view involves a fair amount of algebraic manipulations to proceed from one step to another (and the

augmented matrix formulation is unnatural using this approach), the elimination point of view involves dealing with adding multiples of vectors to each other in order to form the next

augmented matrix (which is often done on the side).

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.

DKPCA fault diagnosis method is that each of the observation variables is expanded by H observations in front, the

augmented matrix containing the first S time observations is constructed [12], and

augmented matrix is as follows:

p,j], which turns out to be the

augmented matrix [[V.

Assuming the new

augmented matrix X(s) is mapped nonlinearly into a high dimensional feature space [PHI] : [R.

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.

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].

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