18 
ME.  A.  CAYLEY  ON  THE  THEOEY  OF  MATEICES. 
nant,  formed  out  of  the  matrix  diminished  by  the  matrix  considered  as  a single  quantity 
involving  the  matrix  unity,  will  be  equal  to  zero.  The  theorem  shows  that  ever)' 
rational  and  integral  function  (or  indeed  every  rational  function)  of  a matrix  may  be 
considered  as  a rational  and  integral  function,  the  degree  of  which  is  at  most  equal  to 
that  of  the  matrix,  less  unity ; it  even  shows  that  in  a sense,  the  same  is  true  with  respect 
to  any  algebraical  function  whatever  of  a matrix.  One  of  the  applications  of  the 
theorem  is  the  finding  of  the  general  expression  of  the  matrices  which  are  convertible 
with  a given  matrix.  The  theory  of  rectangular  matrices  appears  much  less  important 
than  that  of  square  matrices,  and  I have  not  entered  into  it  further  than  by  showing 
how  some  of  the  notions  applicable  to  these  may  be  extended  to  rectangular  matrices. 
1.  For  conciseness,  the  matrices  written  down  at  full  length  uill  in  general  be  of  the 
order  3,  but  it  is  to  be  understood  that  the  definitions,  reasonings,  and  conclusions 
apply  to  matrices  of  any  degree  whatever.  And  when  two  or  more  matrices  are  spoken 
of  in  connexion  with  each  other,  it  is  always  implied  (unless  the  contrary  is  expressed) 
that  the  matrices  are  of  the  same  order. 
2.  The  rrotation 
( « , ^ , c Jx,  y,z) 
a',  b' , c' 
a",  b”,  c" 
represents  the  set  of  liirear  functions 
({a,  b,  cX^,  y,  z),  («',  b\  c'X^,  y,  z),  (a",  b\  c'Xx,  y,  z)), 
so  that  calling  these  (X,  Y,  Z),  we  have 
(X,Y,  Z)=(  «,  ^,6- 
a! , b’ , d 
a\  b",  c" 
and,  as  remarked  above,  this  formula  leads  to  most  of  the  fundamerrtal  irotions  hr  the 
theory. 
3.  The  qirarrtities  (X,  Y,  Z)  will  be  identically  zero,  if  all  the  terms  of  the  matrix 
are  zero,  and  we  may  say  that 
C 0,  0,  0 ) 
0,  0,  0 
0,  0,  0 
is  the  matrix  zero. 
Again,  (X,  Y,  Z)  will  be  identically  equal  to  (x,  _y,  z),  if  the  matrix  is 
( 1,  0,  0 ) 
0,  1,  0 
0,  0,  1 
and  this  is  said  to  be  the  matrix  unity.  We  may  of  course,  when  for  distirrctness  it  is 
