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

196

LINEAR TRANSFORMATIONS

[CH. IX

This we prove by comparing the matrices of these products,
as obtained by the application of Theorem 1.
80. Canonical Form of a Linear Transformation. Identical
and Similarity-transformations. A transformation whose matrix
has zero elements everywhere except in the principal diagonal
is said to have the canonicalform (or to be written in the canonical
form):
S: X1 =aiX, X2 =a2X2,..., Xn O=nXnIn such a case we employ the notation S = (ai, a2,..., an).
If the coefficients at, a2,, an, which are called the
multipliers of S, are all equal, we say that S is a similaritytransformation; if they are all equal to unity, S is the identical
transformation or the identity. Denoting the latter by E and
any transformation by A, we have EA =AE=A.
81. Power and Order of a Linear Transformation. Since
the associative law holds for a product, it follows that we may
write A2 for AA, A3 for (AA)A, etc., and call these products
the second, third, etc., powers of A. Moreover, denoting the
inverse of A' by A-", we have AnA-"=A-'An=E, and
A-"=(A-1)'). The index laws hold for positive and negative
integral powers if we interpret A~ as E.
Usually no power of a linear transformation A taken at
random will be the identity. If, however, such a power exists,
we say that A is of finite order, and the lowest power of A which
equals the identity is called the order of A.
EXERCISES
1. Prove that the determinant of the product of two transformations
A and B is equal to the product of the determinants of A and B. Hence
prove that the determinant of A is the reciprocal of that of A-1.
2. Find the inverse of S, ~ 75.
3. Prove that if
A=r [-r B q ps-qr=l,
= S p q
then B is the inverse of A.
4. Construct AB and BA, where
A= a b|c B=. hi