Chapter 10
Matrix and Difference Calculus
The matrix calculus is used in the study of linear systems and systems of differ-
ential equations and occurs in engineering mathematics, physics, statistics, biology,
chemistry and many other scientific applications. The difference calculus is used to
study discrete events.
The Matrix Calculus
A matrix is a rectangular array of numbers or functions and can be expressed in
the form
A=
a 11 a 12 a 13... a 1 j... a 1 n
a 21 a 22 a 23... a 2 j... a 2 n
..
.
..
.
..
.
... ..
.
... ..
.
ai 1 ai 2 ai 3... aij... a in
..
.
..
.
..
.
... ..
.
... ..
.
am 1 am 2 am 3... a mj... a mn
(10.1)
where the quantities aij for i= 1, 2 ,.. ., m and j= 1, 2 ,.. ., n are called the elements
of the matrix. Here the double subscript notation aij is used to denote the element
in the ith row and jth column. A matrix with mrows and ncolumns is called a m
by nmatrix and expressed in the form “ Ais a m×nmatrix”. Matrices are usually
denoted using capital letters and whenever it is necessary to emphasize the elements
and size of the matrix it is sometimes expressed in the form A= (aij)m×n.The rows
of the matrix Aare called row vectors and the columns of the matrix Aare called
column vectors.
For aand bpositive integers, then matrices of the form R= (ra 1 ra 2... raj... ran )
are called n-dimensional row vectors and matrices of the form
C=
c 1 b
c 2 b
..
.
cib
..
.
cmb
=col(c 1 b, c 2 b,... , cib,... , cmb) (10.2)
are called m-dimensional column vectors. The column notation col(c 1 b,... , c mb)is used
to conserve space in typesetting the m-dimensional column vector.
