On the Matrix
Convolution Product and Its Applications
Adem KILICMAN
akilic@fsas.upm.edu.my
Department of Mathematics
University Putra Malaysia
Malaysia
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, VectorOperator, Laplace
transform, Dirac identity matrix, Renewal matrix equation, Matrix
differential equations, Exponential matrix, Correlation matrix.
