ME.  A.  CAYLEY  ON  THE  THEOEY  OF  MATEICES. 
the  equations  1+B=:0,  A+C=:0,  and  we  cannot,  therefore,  from  the  equation 
(]VP— 1)(M  + C)=:0,  deduce  the  equation  1 = 0.  This  is  as  it  should  be,  for  the 
two  conditions  are  not  sufficient,  in  fact  the  equation 
-\-hd-\-c(j,  ab-\-he-\-cli^  ac-\-hf-\-ci  ) = 1 
da-^ed-\-f(j,  dh-\-e^ dc-\-ef-\-fi 
ga-\-]id-\-ig ^ gh  -\-lie-\-ih,  gc-\-hf-{-i'^ 
gives  nine  equations,  which  are  however  satisfied  by  the  following  values,  involving  in 
reality  four  arbitrary  coefficients ; riz.  the  value  of  the  matrix  is 
^■=( 
I a 
a + ^ + y’’ 
— (7  + «)f4V-‘^ 
« + /3  + 7 
— + 
ot  + /3  + y 
-(/3+r)^ 
-(/3  + 7)^ 
«+^+7 ’ 
a + ^+7 
/3 
-(7  + «)- 
r 
« + /3  + 7’ 
« + /3  + 7 
7 
«+^+7 ’ 
U + ^ + y 
so  that  there  are  in  all  four  relations  (and  not  only  two)  between  the  coefficients  of  the 
matrix. 
32.  Instead  of  the  equation  M”— 1 = 0,  which  belongs  to  a periodic  matrix,  it  is  in 
many  cases  more  convenient,  and  it  is  much  the  same  thing  to  consider  an  equation 
M” — k=0,  where  is  a single  quantity.  The  matrix  may  in  this  case  be  said  to  be 
periodic  to  a factor 
33.  Two  matrices  L,  M are  convertible  when  LM=ML.  If  the  matrix  M is  given, 
this  equality  affords  a set  of  linear  equations  between  the  coefficients  of  L equal  in 
number  to  these  coefficients,  but  these  equations  cannot  be  all  independent,  for  it  is 
clear  that  if  L be  any  rational  and  integral  function  of  M (the  coefficients  being  single 
quantities),  then  L will  be  convertible  with  M ; or  what  is  apparently  (but  only  appa- 
rently) more  general,  if  L be  any  algebraical  function  whatever  of  M (the  coefficients 
being  always  single  quantities),  then  L will  be  convertible  with  M.  But  whatever  the 
form  of  the  function  is,  it  may  be  reduced  to  a rational  and  integral  function  of  an  order 
equal  to  that  of  M,  less  unity,  and  we  have  thus  the  general  expression  for  the  matrices 
convertible  with  a given  matrix,  viz.  any  such  matrix  is  a rational  and  integral  function 
(the  coefficients  being  single  quantities)  of  the  given  matrix,  the  order  being  that  of  tlie 
given  matrix,  less  unity.  In  particular,  the  general  form  of  the  matrix  L convertible 
with  a given  matrix  M of  the  order  2,  is  L=o5M-l-/3,  or  what  is  the  same  thing,  the 
matrices 
( a,  b ), 
\ c,  d 
( V ) 
I c\  d!  I 
will  be  convertible  \i  d ~ d!  \V  \ d — a—d  \ h \ c. 
