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 , and augmented matrix
is as follows:
p,j], which turns out to be the augmented matrix
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
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  by using a Jacobi-Davidson type method which makes use of the block structure of the augmented matrix