TKANSFOEMATION  OF  A BIPAETITE  QUADEIC  FUNCTION. 
45 
Q ‘ =Qo~l“Q/? 
(tr,  Q)"*=tr.  (Q"')  =Qo— Q^, 
Y 
tr.Y  F=Yo-Y^, 
and  the  condition,  Q *Y= — (tr.  Q)  ‘tr.Y,  becomes 
that  is, 
and  we  have 
or  as  we  may  write  it. 
(Q.+Q,)(Y.+Y,)  = -(Q.-Q,)(Y.-Y,), 
Q.Y.+Q,Y,=0, 
Y.=  -Q-'Q,Y„ 
Y,=  -(i{Q-+tr.Q-'})-‘(J{Q-'-tr.  Q-'})Y„ 
and  thence 
Y=-(i{Q-‘+tr.  Q-‘})-‘(i{Q-‘-tr.Q-‘})Y,+Y^, 
where  Y,  is  an  arbitrary  skew  symmetrical  matrix. 
16.  This  includes  the  before-mentioned  special  cases ; first,  if  Q is  symmetrical,  then 
we  have  simply  Y=Yp  an  arbitrary  skew  symmetrical  matrix,  which  is  right.  Next,  if 
Q is  skew  symmetrical,  then  Y=  — 0“‘Q“‘Y^+Yp  which  can  only  be  finite  for  Yj=0, 
that  is,  we  have  Y=  — 0~‘Q"'0,  and  (the  fii’st  part  of  Y being  always  symmetrical)  this 
represents  an  arbitrary  symmetrical  matrix.  The  mode  in  which  this  happens  will  be 
best  seen  by  an  example.  Suppose 
and  wuite 
then  we  have 
Q-'  = ( A , H+v  ),  tr.Q-'  = ( A , H-v  ) 
B 
H+v,  B 
r,=(  0,  6 ) 
0 I 
T=-(  A,  H )-*(  0,  . )(  0,  ^ ) + ( 0,  0 ) 
1 H5  B 1 j — 0 1 — ^5  0 I I — 0,  0 j 
vQ 
AB-H2 
^,(-B,  H ) + ( 0,0 
_ ( 
tB- 
-vH5 
H,  -A  I -0,  0 
ZTMLj-^  ) + ( O 
I 0 I 
AB-H*  ’ AB-H^"^^ 
AB-li 
■vA9 
AB-H^ 
When  Q is  skew  symmetrical.  A,  B,  H vanish ; but  since  their  ratios  remain  arbitrary, 
we  may  write  «A,  «B,  for  A,  B,  H,  and  assume  ultimately  fc=:0.  Writing  xd  in  the 
place  of  6,  and  then  putting  k=0,  the  matrix  becomes 
