of Edinburgh, Session 1871-72. 
675 
Monday , 1 %th March 1872. 
Professor KELLAND, Vice-President, 
in the Cliair. 
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). 
I d, e, f I 
I 9, i 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 work out the solution. 
The notation and method will be understood from the simple 
case of a matrix of the second order. I write 
Oi, yd = ( <», ® ) 0, y), 
I «, <* I 
to denote the two equations, x x = ax + by, y 1 = cx + dy. This being 
so, putting 
(* 2 > yd = ( «, b ) Oi> yj, = ( «, b ) 2 (*, y)> 
I c, d I I c, d | 
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 ) . 
I 0, 1 | 
The zero matrix is ( 0, 0 ), and when a matrix rs put = 0, this 
I o, 0 | 
means that it is a matrix of the last-mentioned form. 
VOL. VII. 
4 u* 
