COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

394 R. E. Kalman
“hermitian forms”). Hermite’s functor has the special form
{polynomial n of degree n} - (nun symmetric matrix P*}
satisfying the property that
{number of roots of 7~ inside the unit circle} = rank of P,.
(3)
Our main object then is to try to exhibit other functors of this general type.
The motivation for this investigation is especially rich; in addition to
pure and applied mathematics, it stems also from the modern mathematical
theory of control and dynamical systems. For instance, a discussion of
Hermite’s functor in the style of Lyapunov stability theory and control
theory was given in [2].
2. Common factors of polynomials. It is well known that the common
factor of two polynomials can be determined by the Euclidean algorithm.
If we wish to avoid (for numerical reasons) dividing polynomials, then we
can make use of a well-known “functor” of type (1) known as the Euler-
Sylvester determinant ([3], ch. 5, p. 104) which is defined as
{polynomials f, g of degree m, n} -+
{determinant R/, g of an mnXmn matrix}. (4)
Thenf, g are relatively prime (have no common factor of degree =- 0) if and
only if Rf.g 9 0.
The “functor” (4) is rather inefficient from the computational point of
view since RLg is a very large determinant. Moreover, if Rf.* vanishes, so
that f, g have a nontrivial greatest common divisor, it is not a simple
matter to compute this common divisor.
We shall now exhibit a “functor” which is much more efficient for the
above purposes.
Notations: let z = indeterminate, K = arbitrary field, deg f = degree of
polynomial $ We assume that n = deg f > deg g (the special case
deg f = deg g causes very little extra difficulty), and define the “codes”
r 1 0
f=z”+fiz”-l+ . ..fn-F=
I
* (5)
g = glz”-l+ . . . +g,, -+ G = (6)
Finally, we write, as usual, cf, g) for the manic polynomial which is the
greatest common factor off and g.
(2)
Invariant factors and control theory 395
THEOREM. (i)[G, FG, . . . . F+‘G] = g(f). Hence &(g(F)) = g(vi), where
yi are the roots off.
(ii> de cf, d = n-rank [G, FG, . . . , F”-‘G].
(iii) V; g) = g/h, w here
h is the minimal polynomial of the vector G
relative to the matrix F, i.e.,
with deg h = r, r = minimum.
F”-‘GfhlF”-‘-1Gf . . . +h,-,G = 0
Outline of proof. Fact (i) (and therefore the fact that (f, g) = 0 if and
only if rank [G, FG, . . . , F”-lG] = n) was first proved in [4], Lemma 7.
Note that in view of(i) (see [3], ch. 5, p. 107) the “functor”
cf, g) -+ det [G, FG, . . ., F+lG] (7)
is identical with the Euler-Sylvester “functor” Rf.g which is now exhibited
more efficiently using an n X n (rather than mn X mn) matrix. (The number
Rfg is the classical resolvent off, g.)
The matrix
[G, FG, . . ., F”-lG],
which is to be thought of as made up of the (column) vectors G, FG, . . .,
plays an important (and well-known) role in modern control theory under
the names “controllability” and “observability”.
The proof of (ii) and (iii) is a straightforward elaboration of (i), see [5].
The form and proof of this theorem suggests rephrasing the algebraic
situation in module-theoretic terms. Recall (this is now classical) that any
square matrix F (over the field K) induces a K[z]-module over K,, = X
regarded as an abelian group. To do this, we define
scalar product: K[z]X K,, -+ K,,
Given a fixed F (and fixed n), the condition
which is equivalent to
means in module language that
: (f, x) - f(F)x.
rank [G, FG, . . ., Fn-lG] = n, (8)
(.A d = 0, (9)
G generates the module X. (10)
This observation is closely related to the theory of realizations which
(especially for our present purposes) may be regarded as a generalization
of the classical theory of elementary divisors.
3. Theory of realizations. Consider the infinite sequence
Y = {Yk: k = 0, 1, . . .) Yk = pXm matrix over K}.