Associated with any matrix, there are four fundamental subspaces: the column space, row space, (right) null space, and left null space. We describe a single computation that makes readily apparent bases for all four of these subspaces. Proofs of these results rely only on matrix algebra, not properties of dimension. A corollary is the equality of column rank and row rank. Bringing a matrix to r...

Combinatorial problems like flow shop scheduling, travel salesman problem etc. get complicated and are difficult to solve when the problem size increases. To overcome this problem, we present a block-based evolutionary algorithm (BBEA) which will conduct evolutionary operations on a set of blocks instead of genes. BBEA includes the block mining and block recombination approaches. A block mining...

‎‎‎This paper presents a remarkable formula for spectral distance of a given block normal matrix $G_{D_0} = begin{pmatrix}‎ ‎A & B \‎ ‎C & D_0‎ ‎end{pmatrix} $ to set of block normal matrix $G_{D}$ (as same as $G_{D_0}$ except block $D$ which is replaced by block $D_0$)‎, ‎in which $A in mathbb{C}^{ntimes n}$ is invertible‎, ‎$ B in mathbb{C}^{ntimes m}‎, ‎C in mathbb{C}^{mti...

in this note‎, ‎we obtain some singular values inequalities for positive semidefinite matrices by using block matrix technique‎. ‎our results are similar to some inequalities shown by bhatia and kittaneh in [linear algebra appl‎. ‎308 (2000) 203-211] and [linear algebra appl‎. ‎428 (2008) 2177-2191]‎.

General sparse matrix-matrix multiplication (SpGEMM) is a fundamental building block for numerous applications such as algebraic multigrid method (AMG), breadth first search and shortest path problem. Compared to other sparse BLAS routines, an efficient parallel SpGEMM implementation has to handle extra irregularity from three aspects: (1) the number of nonzero entries in the resulting sparse m...

Sparse matrix-vector multiplication is an important computational kernel that tends to perform poorly on modern processors, largely because of its high ratio of memory operations to arithmetic operations. Optimizing this algorithm is difficult, both because of the complexity of memory systems and because the performance is highly dependent on the nonzero structure of the matrix. The Sparsity sy...

Conjugate gradient (CG) methods to solve sparse systems of linear equations play an important role in numerical methods for solving discretized partial diierential equations. The large size and the condition of many technical or physical applications in this area result in the need for eecient par-allelization and preconditioning techniques of the CG method. In particular for very ill-condition...

Given a rectangular matrix with more columns than rows, nd a base of linear combinations of the row vectors such that these contain as many zero entries as possible. This process is called \sparsiication" (of the matrix). A combinatorial search method to solve sparsiication is presented which needs exponentially many arithmetic operations (in terms of the size of the matrix). However, various p...