504
A MEMOIR ON THE AUTOMORPHIC LINEAR TRANSFORMATION
[153
or as we may_ write it,
T 0 = — (№-' + tr. - tr. Or*}YC„
and thence
T = - atil- 1 + tr. - tr. il-ijyr, + T„
where T, is an arbitrary skew symmetrical matrix.
16. This includes the before-mentioned special cases; first, if il is symmetrical,
then we have simply T = T / , an arbitrary skew symmetrical matrix, which is right.
Next, if il is skew symmetrical, then T = — 0 -1 i2 -1 T / + T /} which can only be finite
for T, = 0, that is, we have T = — 0 _1 il _1 0, and (the first part of T being always
symmetrical) this represents an arbitrary symmetrical matrix. The mode in which this
happens will be best seen by an example. Suppose
and write
then we have
n-' = ( A , H+v ), tr. O" 1 = ( A , H-v),
| H-v, B I \h + v, B |
t, = ( o, 0),
1-0, o|
T = -(A, H)~'( 0, v)( 0, 0) + ( 0, 6)
| -H, b | j - v, o ! ! - e, o | - e, o |
v6
AB— H 2
(-B,
H,
H) + ( o, 0)
-a\ I - 0, 0 I
( vB6
AB-H 2
-vHQ
AB-H 2
-e,
-vH6
AB - H 2
+ 0
vA6
AB-H 2
0 )■
o I
When il is skew symmetrical, A, B, H vanish; but since their ratios remain arbitrary,
we may write kA, kB, kH for A, B, H, and assume ultimately re = 0. Writing k6
in the place of 6, and then putting re = 0, the matrix becomes
( vB6 — vB6 )
AB- H 2 ’ AB-H 2
— vH6 vAQ
AB — H 2 ’ AB-H 2
which, inasmuch as A : 6, B : 6, and G : 6 remain arbitrary, represents, as it should do,
an arbitrary symmetrical matrix.