Page 169 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 15.3 Linear Operators

In this section we consider more complicated geometric objects. For the sake of certainty, let’s
denote one of such objects by F. In each basis e , e , e , it is represented by a square 3 × 3 matrix
1
3
2
F of real numbers. Components of this matrix play the same role as coordinates in the case of
i
j
vectors or covectors. Let’s prescribe the following transformation rules to F :
i
j
3 3
q
i
p
i 
j 
F  T S F , ...(1)
q
p
j
p 1 q 1


3 3
i
j 
q 
i
p
F  S T F . ...(2)
q
p
j
p 1 q 1


Exercise 15.5: Using the concept of the inverse matrix T = S prove that formula (2) is derived
–1
from formula (1).
i
If we write matrices F and F , then (1) and (2) can be written as two matrix equalities:
p 
q
j


F  T F S, F  S F T. ...(3)
Definition: A geometric object F in each basis represented by some square matrix F and such
i
j
that components of its matrix F obey transformation rules (1) and (2) under a change of basis is
i
j
called a linear operator.
Let’s take a linear operator F represented by matrix F in some basis e , e , e and take some
i
j
3
1
2
vector x with coordinates x , x , x in the same basis. Using F and x we can construct the
2
j
i
1
3
j
following sum:
3
j
y =  F x . ...(4)
i
i
j
j 1

Index i in the sum (4) is a free index; it can deliberately take any one of three values: i = 1,
i = 2, or i = 3. For each specific value of i we get some specific value of the sum (4). They are
denoted by y , y , y according to (4). Now suppose that we pass to another basis e ,e ,e  and do
1
3
2
 
3
1
2
the same things. As a result we get other three values y ,y ,y  given by formula
1
2
3


3
p
q
y   F x . ...(5)
p  

q
q 1

Relying upon (1) and (2) prove that the three numbers y , y , y and the other three numbers
1
3
2
y ,y ,y  are related as follows:
1
3
2


3 3
j
j
i
j
i
j
y   T y , y   S y . ...(6)


i
i
i 1 i 1


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