相对因子宽度与可S-因子分解矩阵

Relative factor width and S-factorizable matrices

设S是实数集R的一个非空子集,如果存在S上的矩阵B,使得A=BBT,则称A是可S-因子分解的.对于一个实对称矩阵A,如果存在一个最小正整数k以及实矩阵(长方形)V,使得A=VVT,且V的每一列至多只有k个非零元素,则称A的因子宽度为k.利用可S-因子分解矩阵的S-秩以及因子宽度,引入相对因子宽度的定义,给出了一些可{0,1}-因子分解矩阵的相对因子宽度与因子分解之间的关系,最后利用S-秩和相对因子宽度,刻画了一类矩阵.

Let S be a non-void subset of a set of real numbers R. A is called S-factorizable if it can be factorized as A = BB^T with bij∈ S. For a real symmetric matrix A, if there exists a minimum positive integer k and a real rectangular matrix V with A = VV^T, and there are at most k non-zero elements in each column, the factor width of A is k. By use of S-rank and factor width of the S-factorizable matrix A, the concept of relative factor width was introduced, and the relationship between the relative factor width and factorization of some{0, 1 }-factorizable matrices was given. Finally, some matrices were characterized with S-rank and relative factor width.