1. ELEMENTAR Y PROPERTIE S O F HERMITIA N FORM S 5
U o f V, Lemma 1.5(3) together wit h our proof o f (1) guarantees a n isomorphism /
of (V , p) ont o (X , £ ) 0 (H m, rjm) such tha t Uf C Jm. Thi s prove s (3) .
It shoul d b e note d tha t (V , ip) ma y contai n variou s anisotropi c subspace s o f
dimension large r tha n dim(X) . Fo r example , le t (V , if) = (5 , a) 0 (5 , a) wit h
nondegenerate a. Le t U be the subspac e o f V = S 0 S consistin g o f all the vector s
(5, s) wit h s G S. Clearl y U i s totall y (^-isotropic , an d henc e b y Lemm a 1.5(3),
(V, (p) is isomorphi c t o (H
n
, ry
n
) wit h n = dim(5) . Thi s mean s tha t (H
n
, rj
n
) ca n
contain a n arbitrary nondegenerat e (5 , a) o f dimension n. Th e spac e S ma y hav e
an anisotropi c subspace .
1.7. Le t u s no w expres s variou s thing s b y matrices . T o simplif y ou r notatio n
we put x* = t xp fo r ever y matri x x wit h entrie s i n K, an d pu t als o x = (x*) _1
if x i s square an d invertible . Now , give n (V , (p) with degenerat e o r nondegenerat e
p, tak e a if-basi s {e
i}7}^l
o f V. Fo r x =
Y17=i€iei
^ ^
w
^
n
& ^ ^ ^
et x
°
be th e elemen t o f K\ whos e component s ar e £1, ... , £n. The n w e ca n defin e a n
isomorphism a \— a$ o f GL(V) ont o GL n(K) b y (xa)o = £o2o - Lety? o b e th e
n x n-matri x [p(ei, ej)]
{ =l
The n y? Q = £P o an d /?(# , y) = XQipoy^; moreover , i f
(p is nondegenerate, th e ma p a 1—• a o give s a n isomorphis m o f G* ont o th e grou p
(1.7.1) G(p
0
) = {pe GL
n
(K) I /fyo/T = y o } .
We call (po the matrix that represents (p with respect to {ei}™ =1, or simply a matrix
representing p whe n th e basi s i s no t specified. Hereafter , w e shal l ofte n emplo y
this matri x expression, an d w e use the same letter fo r a n £-hermitia n for m an d th e
matrix representin g i t whe n th e basi s i s fixed. I f w e change th e basis , the n ipo is
changed int o £po£* wit h £ G GLn(K). W e easily se e tha t
(1.7.2) G(po) = { 0 G GLn(K) \ (3*Po ^ = ¥ o ' }
Notice tha t 7?
r
o f §1.4 is represented b y the matri x
(1.7.3)
V r
= [ l * - ]
with respec t t o th e standar d basi s o f H
r
= K\ r.
1.8. Lemma , Suppose that p ^ id ^ or e = 1; suppose also that the character-
istic of K is not 2. Then every degenerate or nondegenerate (p can be represented
by a diagonal matrix.
Proof. This i s trivial i f ip = 0 or n = 1. We prove the genera l cas e by inductio n
on n . Suppos e ip ^ 0 ; tak e x, y G V s o that p(# , 2/)7^ 0. Pu t 2 = p(x, y)~ xx i f
£ = 1. I f p 7^ idx an d £ = —1, tak e c G X s o tha t c p ^ c an d put 2 = op(x, y)~ lx.
In eithe r cas e we have
p(y + z,y + z)- ip{y, y) - ip(z, z) = p(z, y) + ep(z, y) p ^ 0 .
This mean s tha t w e ca n fin d a n elemen t w o f V suc h tha t (p(w, w) ^ 0 . Pu t
U = { v G V I p(v, w) = 0 } . The n V =
JKI U
0 (7 . (I n fact , give n x G V , pu t
-u = x (p(x, w)p(w, w)~ 1w. The n u £ U. Also, we easily see that KwDu = {0}. )
Applying inductio n t o th e restrictio n o f ip to U, we obtain ou r lemma .
We now presen t a generalization o f the classica l Cayle y transformation .
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