30 
ME.  A.  CAYLEY  ON  THE  THEOEY  OF  MATEICES. 
34.  Two  matrices  L,  M are  skew  convertible  when  LM= — ML;  this  is  a relation 
much  less  important  than  ordinary  convertibility,  for  it  is  to  be  noticed  that  we  cannot 
in  general  find  a matrix  L skew  convertible  with  a given  matrix  ^I.  In  fact,  con- 
sidering M as  given,  the  equality  affords  a set  of  linear  equations  between  the  coeffi- 
cients of  L equal  in  number  to  these  coefficients ; and  in  this  case  the  equations  are 
independent,  and  we  may  ehminate  all  the  coefficients  of  L,  and  we  thus  arrive  at  a 
relation  which  must  be  satisfied  by  the  coefficients  of  the  given  matrix  M.  Thus,  sup- 
pose the  matrices 
( a,  ^ ) , ( V ) 
are  skew  convertible,  we  have 
( ^ V 
I c,  d c',  d! 
( a',  V )(  a,  I 
c',  d!  c,  d 
) = ( ad-\-hd , ah'-\-bd'  ), 
ca! -\-dc\  cV -\-dd!\ 
^ = aa!-\-h'c,  db-\-b'd  ), 
I c'a-\-d'c,  c'b-\-d'd  | 
and  the  conditions  of  skew  convertibility  are 
2aa'-\-bc'-}-b'c  =0 
b'(a-\-d)-\-b(a'  -\-d’)  = 0 
(/(a-{~d)-{-c{a'-{-d')=0 
2dd’+bd+b'c  =0 
Eliminating  a',  b',  c',  d',  the  relation  between  a,  b,  c,  d is 
which  is 
b , a-\-d,  . b 
c,  . a-\-d,  c 
c , b ^ 2d 
{a-^-dyi^ad—bc)—^. 
Excluding  from  consideration  the  case  ad—bc~^^  which  would  imply  that  the  inatiix 
was  indeterminate,  we  have  a-{-d=^.  The  resulting  system  of  conditions  then  is 
a-\-d=^,  a!-\-d!  = (),  aa!  -\-bd  -\-b'c-\-dd!  — 
the  first  two  of  which  imply  that  the  matrices  are  respectively  periodic  of  the  second 
order  to  a factor  pres. 
35.  It  may  be  noticed  that  if  the  compound  matrices  LM  and  ML  are  similar,  they 
are  either  equal  or  else  opposite ; that  is,  the  matrices  L,  M are  either  coinertible  or 
skew  convertible. 
