CHARACTERISTIC EQUATION 
[chap. VII 
latrix R x 
p row of 
1* x > since 
/ with a 
Again, 
vithout a 
nents are 
A, and is 
bordered 
) x with a 
ad hence 
■j by a kj . 
nmutative 
: socialise. 
=xz f and 
he group 
r in Lie’s 
99 
theory of continuous groups. This was the origin of 
the term “reciprocal algebras” (§ 12). 
59. Characteristic determinant and equation of a 
matrix. Let x be an w-rowed square matrix with 
elements in a field F. Let co be an indeterminate. Write 
(15) /(co) = | x-wl | 
for the determinant of matrix x — col. Thus /(co) is a 
polynomial of degree n in co with coefficients in F. 
It was proved at the end of § 3 that 
(16) (x—co/)adj. {x—. 
Each member may be expressed as a polynomial in co 
whose coefficients are matrices independent of co. Hence 
the coefficients of like powers of co are equal. Thus, if 
m is any matrix commutative with x, the corresponding 
polynomials obtained by replacing co by w are identical, 
and the same is true of the members of (16). But if 
we take m = x and replace co by x in the left member of 
(16), we obtain the matrix o. Hence f(x)l = 0. 
We shall call /(co) and /(co) = o the characteristic 
determinant and characteristic equation of matrix x. 
Theorem. Any matrix x is a root of its characteristic 
equation. It is understood that when co is replaced by x 
the constant term c of /(co) is replaced by cl. 
60. Characteristic matrices, determinants, and equa 
tions of an element of an algebra. Let g(co) be any 
polynomial with coefficients in F which has a constant 
term c^o only when the associative algebra A over F 
has a modulus e and then the corresponding polynomial 
g(x) in the element x of A has the term ce. Then the 
first and second matrices of g(x) are 
(by) Fg{x)—g(F- x )) Sg(x) ~x) •