An integral representation of the determinant of a matrix and its applications

The Hadamard determinant theorem states that the ratio of the determinant of a square matrix over the complex field to the product of its main diagonal elements is less than or equal to one for a positive definite Hermitian matrix. An integral representation of this ratio for both positive definite Hermitian matrix and diagonally dominant real matrix is given in this paper. Using this new identity, an alternative proof of the famous Hadamard determinant theorem is discussed. In addition, a lower bound of determinant in terms of the product of the main diagonal elements is given. Finally, a numerical algorithm fundamentally different from current approaches in literature is also proposed for the computation of the determinant of a small and dense matrix. Numerical experiments indicate that this new approach is robust.