On the Matrix Convolution Product and Its Applications

Abstract

In addition to the usual matrix multiplication; recently there has been renewed interest in another kind matrix multiplication that is also known as convolution product and it is very useful in applications of matrix equations and matrix differential equations in matrix theory, engineering and many other subjects. In this work, we study several properties (equalities and inequalities) of the convolution product of matrices .We define the Dirac identity matrix and its properties which behaves like a group identity element under the convolution matrix operation. We also derive several elementary properties of the matrix convolution product. The connections between the usual product and the convolution product of matrices are established. Further some applications of the convolution matrix product are also provided. These applications involve with renewal matrix equation and non homogeneous matrix differential equations. Keywords: Convolution product of functions, Convolution product of matrices, Kronecker product (sum) of matrices, Matrix norm, Vector-Operator, Laplace transform, Dirac identity matrix, Renewal matrix equation, Matrix differential equations, Exponential matrix, Correlation matrix.