Study On The Properties Of Some Special Divisible Matrices And Its Determination

The study of infinitely divisible matrices not only enriches the content of matrix theory,but also has applications in the branches of complex analysis,Fourier analysis and probability theory.The MAX matrices and MIN matrices serve as special matrices,which provide more examples for infinite divisible matrices,and their properties are also worth studying.Based on the above research purposes,this paper mainly studies some problems about MAX matrices and MIN matrices.The specific work is as follows:Firstly,the MAX matrix M?n?and MIN matrix MIN?n?are generalized to the block MAX matrix M?n?and block MIN matrix M?n?and the relationship between them is given by subtranspose operations:^TM_?n?is a block MIN matrix,and related issues about M?n?are equivalently converted to issues about M?n?by proving that M?n??^TM_?n?.Secondly,the determinant,inverse and characteristic polynomials of M?n?are obtained by using the given decomposing formula which based on the relationship of M?n?cotracting on the block symmetric triangular matrix;using the characteristics of the Chebyshev polynomial and the symmetric tridiagonal matrix determinant completely solves the general expressions of characteristic polynomials for arithmetic MAX matrix;using the Chebyshev polynomials to find and construct eigenvalues and eigenvectors of the arithmetic MAX matrix M?n,2?and M,and further give the eigendecomposition and power of the matrix.Next,the necessary and sufficient conditions for positive definiteness and infinite divisibility of the MAX matrix and MIN matrix are given.As corresponding inferences,other special infinitely divisible matrices are obtained;the connection between the MAX matrices and M-matrices is studied:the positive definite MAX matrices are closed under Hadamard multiplication as the inverse M-matrices;the totally nonnegativity of the MAX matrices is also discussed;a necessary condition for the infinite divisibility of a general matrix is given and the infinite divisibility of a general symmetric tridiagonal matrix is discussed.Finally,the article work is summarized and several problems and conjectures are put forward.