Page 71 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes  a d
and T   = , T () =  c , if c  0
1
1

c 
From the above discussion, we conclude that inverse of a bilinear transformation is bilinear.
d a
The points z =  (w = ) and z = (w = ) are called critical points.
c c
Theorem

Composition (or resultant or product) of two bilinear transformations is a bilinear transformation.
Proof. We consider the bilinear transformations

az b

w = cz d , ad  bc  0 (1)

a w b

and w = c w d 1 1 , a d  b c  0 (2)
1
1
1
1 1
1

1
Putting the value of w from (1) in (2), we get

a 1   az b   b 1




w =  cz d   (a a b c)z (b d a b)
1
1
1
1

1



c 1    az b   d 1 (c a d c)z (d d c b)
1
1
1
1
cz d 

Taking A = a a + b c, B = b d + a b,
1
1
1
1
C = c a + d c, D = d d + c b, we get
1
1
1
1
Az B

w = Cz D

1
Also AD  BC = (a a + b c) (d d + c b)  (b d + a b) (c a + d c)
1
1
1
1
1
1
1
1
= (a ad d + a ac b + b cd d + b cc b)  (b dc a + b d d c + a bc a + a bd c)
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
= a ad d + b bc c  b dc a  a bd c
1
1
1
1
1
1
1
1
= ad(a d  b c )  bc(a d  b c )
1
1
1
1
1 1
1 1
= (ad  bc) (a d  b c )  0
1
1
1 1
Az B

Thus w = Cz D , AD  BC  0
1

is a bilinear transformation.
This bilinear transformation is called the resultant (or product or composition) of the bilinear
transformations (1) and (2).
The above property is also expressed by saying that bilinear transformations form a group.
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