关键词: 整数矩阵,整数左右可逆性,Moore-Penrose逆,最小二乘法

Abstract: We considered the discrete data observation model based on integer matrix like boolean observation matrix,or non-negative observation matrix in history.We proposed the problem on the integer decomposition for any integer observation matrix.The problem is equivalent to solving one classical Diophantine linear system in nature,but the only condition is that the base matrix is originated from the decomposed matrix part.Then we provided a new method based on left-right inverse to find the integer inverse of the base matrix.Because the base matrix choice is an option designing in engineering,it is convenient to consider the base matrixes only of possessing integer inverse.Additionally,it is known that any non-full rank matrix can be decomposed into the product of two full rank matrixes.So we explored some sufficient conditions on the existence of integer left-right inverse only for full-rank rectangular matrix,and also designed the integer inverse algorithm in computation.Our proposed integer inverse constructor can find integer inverse under some conditions while the constructor based on the least squares method fails.Lastly our method merged with the least squares method will become a complete solution to integer decomposition of integer observation matrix.

Key words: Integer matrix,Integer left-right invertibility,Moore-Penrose inverse,Least squares method