of Edinburgh, Session 1871 - 72 . 
67 5 
Monday, 18 tli March 1872. 
Professor KELLAND, Vice-President, 
in the Chair. 
The following Communications were read :— 
1. On the Extraction of the Square Root of a Matrix of 
the Third Order. By Professor Cayley. 
Professor Tait has considered the question of finding the square 
root of a strain, or what is the same thing, that of a matrix of the 
third order— 
( a, b, c ). 
d, e, / 
9 , i 
A mode of doing this is indicated in my “ Memoir on the Theory 
of Matrices” [Phil. Trans., 1858, pp. 17-37), and it is interesting 
to w.ork out the solution. 
The notation and method will be understood from the simple 
case of a matrix of the second order. I write 
Oi, y x ) = ( a, b ) (x, y), 
I c, d | 
to denote the two equations, x x = ax + by, y x = cx + dy. This being 
so, putting 
( x - 2 , V-i) = ( ' <*>, b ) (x x , y x ), = ( a, b ) 2 (x, y), 
I c, d I I c, d I 
we arrive at the value of the squared matrix, viz., 
( a, b ) 2 = ( a 2 + be, b(a + d) ) , 
| c, d | | c(a + d), d' 2 + be \ 
and we have similarly the third, fourth, and higher powers of a 
matrix. The zero power is the matrix unity, = ( 1, 0 ) . 
11 0, 1 I 
The zero matrix is ( 0, 0 ), and when a matrix is put = 0, this 
| 0, 0 I 
means that it is a matrix of the last-mentioned form. 
VOL. VII. 
4 u* 
