A NEW THEORY OF LINEAR TRANSFORMATIONS AND
PAIRS OF BILINEAR FORMS
BY PROFESSOR L. E.
DICKSON,
University of Chicago, Chicago, Illinois,
U.S.A.
It is customary to develop the theory of pairs of bilinear forms having
the matrices M and N, and by considering the special case in which N is the
identity (or unit) matrix I to deduce the theory of the canonical form of a linear
transformation T. We here proceed in reverse order and first develop independently a simple theory of linear transformation and later deduce the theory
of equivalence of pairs of matrices and hence of pairs of bilinear forms. We
avoid the introduction of irrationalities and employ only rational processes, so
that our theory holds for any given field (or domain of rationality). We obtain
a simple interpretation of invariant factors.
Moreover, we avoid the consideration of matrices whose elements are any
polynomials in a variable X as well as elementary transformations of matrices.
Start with any given linear transformation
T:
f/ = a,. 1 £ 1 +. ..+ain£n,
(i = l,
...tn)t
whose determinant |a#| may or may not be zero. We shall say that T replaces
ii by £/. Let Xi be any linear homogeneous function of £1, . . . , £ « and let T
replace it by x\.
If x\ is not the product of x\ by a constant, write x2 for x\
and consider similarly the function x'2 by which T replaces x2. If x'2 is not a
linear combination of x± and x2, we write x3 for x'2. In this manner, we obtain
a chain of linearly independent linear functions x\, x2, . . ., xa of the & such that
T implies
(1)
Xi=x2,
x2' = xz, ...,x'a-i
—xa, x'a = lin. fune. X\, . . ., xa.
Select Xi so that it is the leader of such a chain of maximal length a. If n — a,
(1) is the desired canonical form of T.
If n>a, let rjx be any linear homogeneous function of the £ which is linearly
independent of Xi, . . ., xa. As before, T implies
^7i/ = ^2, Vf2 = V3^ • • -, v'b-i = Vbi *7&' = lin.
Iun
C #1» • • •» xa, Vu • • -, Vb-
It is possible to choose a linear function p of X\, . . ., xa so that, if T replaces p
by p', p' by p", etc., and if we introduce the new variables
yi = m+p, y2 = v2+pf, . . . , 3 ; ô : = ^ + P ( 6 ~ 1 \
362
L. E. DICKSON
then T implies
(2)
3;/i = 3;2, 3;/2 = 3;3, • . . , y'b-i^Jb,
y'b = hn. fune. yu
...,
yb,
where the last function involves no one of xi, . . . , xa. Let 771 and hence yi be
chosen so that b is the maximal length of a chain whose leader yi is linearly
independent of Xi, . . . , xa. If n = a+b, (1) and (2) together give the desired
canonical form of T.
But if n>a+b, there exists a third chain zu . . . , zc such that
(3)
z'i=z2, zf2 = z3, . . . , z'c_x-zc,
s'c = lin. fune. zu ...,
zc,
such that c is the maximal length of a chain whose leader zx is linearly independent
of xx, . . ., xa, yu . . ., yb.
This process proves that T has a rational canonical form* C composed of
partial transformations (1), (2), (3), etc., and such that a, b, c, . . . have the
maximal properties described.
If we subtract X from each diagonal element of the matrix A of a linear
transformation T, we obtain the characteristic matrix (or X-matrix) of A or T.
The determinant of this X-matrix is called the characteristic determinant of
A or T.
Let cn, . . ., ak be the characteristic determinants of the partial transformations on the variables of the first, . . ., &th chains of a canonical transformation C. Then a,- is divisible by ai+i for every i. Moreover, the g.c.d.
of all (n — i)-rowed determinants of the X-matrix M of C is the product
a; + i ai+2. . .ak if i<k, but is unity if i^k.
By choice of the signs in
a i = ±In,
a 2 = zkln-v
. . . , ak— dtzIn-k-\-v
we obtain polynomials /„, . . . in X in which the coefficient of the highest power
of X is unity. Write In_k=l,
. . . , Ii—l. Then Ij is the well-known jth invariant
factor of M. By the above results, Ij divides Ij+i, and
(4)
h=
*jP->
(j=l,...,n),
where Gj is the g.c.d. of all/-rowed determinants of M chosen so that the coefficient of the highest power of X in Gj is unity, while G0—l. By (4), the G's
uniquely determine the I's, and conversely.
It is readily proved that Gj does not change when we introduce new
variables into a linear transformation T. Hence T has the same invariant
factors Ij as its canonical form C. The invariant factors other than unity of
any linear transformation T are therefore (apart from signs) the characteristic
determinants of the partial transformations of the canonical form C of T.
Two linear transformations (or matrices) 5 and T are called similar if
there exists a matrix B whose determinant is not zero such that BSB~l=*T.
*It is easy to deduce the classical canonical form and the supplement relating to the irrationalities appearing in the new variables.
PAIRS OF BILINEAR FORMS
363
Then T may be derived from 5 by the introduction of new variables defined
by a linear transformation of matrix B. Hence two linear transformations
(or matrices) are similar if and only if their X-matrices have the same invariant
factors.
Two pairs of n-rowed square matrices M, N and R, S (or bilinear forms
with those matrices) are called equivalent if and only if there exist ?z-rowed
matrices P and Q whose determinants are not zero such that
(5)
PMQ = R,
PNQ = S.
This is true if and only if M—\N and R—\S have t,he same invariant factors,
provided the determinants of N and 5 are not zero. First, if (5) hold, then
MN~X = J is similar to RS~l = K since
PJP-^PMQQ^N^P-^RS-^K,
so that the X-matrices of / and K have the same invariant factors, nd the samea
is true of
(J-\I)N
= M-\N,
(K-\I)S
= R-\S.
Conversely, if the latter have the same invariant factors, J and K are similar,
so that there exists a matrix P whose determinant is not zero such that
PJP~l = K. Then
P(J-\I)P~l
= K-\I,
P(MN~1-\I)P-1
=
RS^1-\I,
P(M-\N)N~1P~1=(R-\S)S^\
Writing Q for N-'p-'S,
we get P(M-\N)Q
= R-\S
or (5).
All of the preceding results were obtained by rational processes and hence
hold for any field.
Next consider a pair of bilinear forms 0 and \p in the variables Xi, . . ., xr,
yi, . .., ys for the singular case in which either r^s or else r = s and the determinant of u$+v\f/ is zero identically in u, v. The theory is due to Kronecker
(Sitzungsber. Akad. Berlin, 1890, 1225-37), who introduced irrationalities in
order to employ the canonical form of Weierstrass for a pair of bilinear forms
in the non-singular case. For the latter we may however employ the rational
canonical form which follows from the foregoing theory. Hence we may replace
Kronecker's theory by a purely rational one, valid for any field.
The new theory of which this is a bare outline will appear in a book on
modern theories of algebra to be published by Sanborn & Co. of Chicago.