442
A SUPPLEMENTARY MEMOIR ON THE THEORY OF MATRICES. [357
and
(T + il) (T - il)“ 1 = ( 1-2 d, -2 h, -21, -2p ),
— 2c, 1 — 2g, — 2k, — 2 o
2b, 2/, 1+2\j, 2 n
2a, 2e, 2 i, 1 + 2m
so that these matrices are composed of terms which, except as to the signs, are the
same in each.
11. Now in general if
© = (
a ,
/3 ,
7 >
S X
a' >
/3' ,
/
7 »
S'
/3",
//
7 »
S"
then it is easy to see that
a'",
/3"',
///
7 >
S'"
il- 1 ©il = (
S'",
///
7 >
-r,
- a
8",
//
7 >
-/3",
— a"
-v,
/
“7 »
/3' ,
a'
-s ,
“7 »
/3 ,
a
and hence, from the foregoing values of (T — il) (T + il) -1 and (T + il)(il — T) x ,
find
n = -il- 1 (T-n)(T + il)- 1 il = ( -1 +2d,
2c,
-2b,
— 2a
2h,
-i + 2g,
-2/,
- 2e
21,
2k,
CM 5
1
r—1
1
- 2 i
2p,
2 o,
— 2 n,
— 1 — 2m
n- 1 = -il" 1 (T + il)(T-il)- 1 il = ( — 1 — 2m,
-2 i,
2e,
2a
— 2 n,
1
t-i
1
2/,
2b
-2 o,
-2k,
-1+2g,
2c
— 2p,
-21,
+ 2 h,
-1 + 2 d
we
this shows that the matrix II for the automorphic transformation of the function
xw' + yz — zy' — wx' is such that writing
n = ( A,
B,
c,
D ) we have II“ 1 = (
P,
L,
-H,
-D ),
E,
F,
G,
H
0,
K,
-G,
-C
I,
J,
K,
L
-E,
-J,
F,
B
M,
N,
0,
P
-M,
-I,
E,
A
which is the theorem in question.