Complementary Basic Matrices Of The Symmetrized Algebra Of Max-plus Algebra

Matrix theory is an important branch of mathematics which has a long history and rich content. Matrix theory is a basic mathematical tool in mathematics and other fields.However, with the increasingly mature and the development of the matrix theory,considerable content are updated in recent years, especially on special matrix. So the study of new special matrix has been closely watched by some scholars.We introduce the equivalence relation ? on R2 max, We set S = R2max/? and name it the symmetrized algebra of Rmax. The paper mainly studies the complementary basic matrices(Complementary basic matrices is short for CB-matrices.) in S. It first introduces the concepts of the intrinsic products and proves the Laplace’s Theorem in S. From this base there can get that: The determinants of CB-matrices are the same and for any one nonzero term in the determinant of one CB-matrix, there exists an equal term in the determinant of the other CB-matrix and vice versa. So if there exist two determinants whose terms are one-to-one corresponding, there must also exist a couple of one-to-one corresponding permutations, then for a given permutation which determines the nonzero term of the determinant of one CB-matrix, there can get a method to find a permutation who determines the same nonzero term in the determinant of the other CB-matrix. In the end, we conclude that the max permanent of the result is equal to the product of simpler max permanents in S.