Chapter 2
Basic Matrix Algebra
Linear Transformations and Matrix Manipulation
The geometrical laws in terms of the first order
theory may be stated in the form of linear
transformations which in turn are
most easily treated by matrix
algebra. In particular we may represent the
physical situation in terms of two linear equations.
The two variables we shall call y and
.
The physical meaning attached to these two variables will
be stated later.
We will be able to show that two other variables
y '
and
'
are related to
y and
through two linear equations of the form:

(1) 

(2) 
where
b_{11}, b_{12},
b_{21}, and b_{22}
are constants, which we will later show to be
characteristic of a given optical system.
These constants may be written as the elements of a
2 x 2 matrix,
The linear equations
(1) and (2) can be written as the matrix equation

(3) 
In fact, equations (1) and (2) constitute
a definition of equation (3).
Now suppose that a further linear
transformation is known which connects
the variables y '' and
'' say, to y
' and
' such that

(4) 

(5) 
or in matrix notation

(6) 
We can discover the equations which relate
y and
to y '' and
'' by
substituting equations,(4) and (5) into (1) and (2). Then we have
or

(7) 

(8) 
which is a new transformation of the form

(9) 

(10) 
In matrix notation,

(11) 
where the "a" coefficients are defined by
comparing equations (9) and (10) with
equations (7) and (8).
The matrix formulation involves the substitution
of the expression for the column matrix
of equation (6) into equation (3) to give

(12) 
Thus comparing equations (11) and (12) we have

(13) 
This essentially defines the
usual rule of matrix multiplication given by the
coefficients of equation (7) and (8) as:
or more compactly,

(14) 
The matrix equation may be written as
Exercises

Show that

Show that

Show that in general

For what particular matrix B does

Show that

Show that
 Show that the multiplication by the matrix
and any matrix
always leaves the element
a_{21}
unchanged.

Compute the product matrix
Determinant of a Matrix
We shall have occasion to make use of one
important property of matrices
which involves the determinant of a matrix.
The determinant of the matrix [A] is written as
and is defined as the number
(a_{11} a_{22} 
a_{21} a_{12}).
The theorem that we are interested in states that the determinant
of the product of a number of matrices is equal to the product of the
determinants of each of the matrices forming the product.
We leave the demonstration of this theorem to the exercises below.
Exercises

Show that

Show that

Show that if
[C] = [B] [A] then B A =
A B = C.

Extend the result of exercise 11 to show that for
N matrices such that
[P] = [A_{1}] [A_{2}] . . .
[A_{n}],
P = A_{1} A_{2} . . .
A_{n}.

Show that the result of exercise 12 holds
for the product matrix of
exercise 8 above.
We are not going to be concerned with other properties of matrices.
The interested student may find these in any
standard text on matrix algebra.