8 Elementary operations and the rank of a matrix 25 8.1 Gauss transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 8.2 Elementary row operations . . . . . . . . . . . . . . . . . . . . . . . . . 26 8.3 The augmented matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 8.4 Row reducing a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 8.5 Elem...

in this paper, we consider a class of column-weight two quasi-cyclic low-density paritycheck codes in which the girth can be large enough, as an arbitrary multiple of 8. then we devote a convolutional form to these codes, such that their generator matrix can be obtained by elementary row and column operations on the parity-check matrix. finally, we show that the free distance of the convolution...

The row-by-row frontal method may be used to solve general large sparse linear systems of equations. By partitioning the matrix into (nearly) independent blocks and applying the frontal method to each block, a coarse-grained parallel frontal algorithm is obtained. The success of this approach depends on preordering the matrix. This can be done in two stages, (1) order the matrix to bordered blo...

Abstract The group $G = \textrm{GL}_r(k) \times (k^\times )^n$ acts on $\textbf{A}^{r n}$, the space of $r$-by-$n$ matrices: $\textrm{GL}_r(k)$ by row operations and $(k^\times scales columns. A matrix orbit closure is Zariski a point for this action. We prove that class such an in $G$-equivariant $K$-theory n}$ determined matroid generic point. present two formulas class. key to proof show clo...

For vectors X, <span style="fon...

Unlike the method without pivoting, Gaussian elimination with partial pivoting consecutively applies row permutation to matrix A in order to avoid possible akk diagonal entries of matrix A being equal to zero. Gaussian elimination with partial pivoting solves the matrix equation Ax = b decomposing matrix A into a lower L and upper U triangular matrices such that PA = LU, where P is a row permut...

A matrix A of size m œ n containing items from a totally ordered universe is termed monotone if, for every i, j, 1 ‹ i < j ‹ m, the minimum value in row j lies below or to the right of the minimum in row i. Monotone matrices, and variations thereof, are known to have many important applications. In particular, the problem of computing the row minima of a monotone matrix is of import in image pr...

The eccentricity matrix E(G) of a connected graph G is obtained from the distance by keeping largest nonzero entries in each row and column, leaving zeros remaining ones. eigenvalues are E-eigenvalues G. In this article, we find inertia matrices trees. Interestingly, any tree on more than 4 vertices with odd diameter has exactly two positive negative (irrespective structure tree). Also, show th...

We focus on row sampling based approximations for matrix algorithms, in particular matrix multipication, sparse matrix reconstruction, and l2 regression. For A ∈ R (m points in d ≪ m dimensions), and appropriate row-sampling probabilities, which typically depend on the norms of the rows of the m × d left singular matrix of A (the leverage scores), we give row-sampling algorithms with linear (up...