Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 891

CHARACTERISTIC EQUATION

205

89. Characteristic Equation. If we add -0 to each of the
elements in the principal diagonal of the matrix of a linear
transformation A = [as] and equate the resulting determinant
to zero, we have an equation in 0 which is called the characteristic equation of A:
all — a12. ain
(6) 21 a a22 -... a2 0
anl an2 ~ ~ ann- 0
THEOREM 3. If T and A be linear transformations, the roots
of the characteristic equation of A are the same as those of T- A T.
To prove this theorem, let us put T-A T =B = [b,], whose
characteristic equation is
bll-0... bln
(7)...... =0.
bnl. ~. bnn- 0
Regarding 0 as a variable temporarily, we denote the transformations whose matrices are the left-hand members of (6) and
(7) by A - and B-0. Then, since T-1AT=B, and T-1ST=S,
where S is the similarity-transformation (0,,..., 0), we
may readily prove that T-1(A-0)T=B-0. Hence, if the
determinants of T, A -6 and B- 0 be denoted by p, q, r, we have
(cf. Ex. 1, ~81) p-qp=r, so that q=r. Accordingly, the
coefficients of the various powers of 0 in q and r are equal, and
the theorem follows.
The sum of the characteristic roots of A is called the characteristic of A. It is equal to the sum of the elements in the
principal diagonal of A, namely al +a22-+... +an,.
EXERCISES
1. Prove that if S, ~ 88, is a similarity-transformation, then S1=S.
Prove also that if S and T both have the canonical form, then S1=S.
2. Prove (5) in the case of two variables directly by multiplying out
the right-hand member (cf. ~~ 77, 78).
3. Find the characteristic roots and characteristic of a transformation written in canonical form; also of the transformation (1), ~ 75.