COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

396 R. E. Kalman Invariant factors and control theory 397
We say that Y has a finite-dimensional realization if and only if there exist
matrices F, G, H over K such that
Yk=HFkG, k=O,l,.... (11)
The matrix F is required to be nX n (then H is pXn and G is nXm). We
say that the realization is minimal if and only if n is the smallest integer for
which (11) can be satisfied. The following theorem is fundamental:
Zf Fy and & belong to minimal realizations or the same Y, then they are
similar.
A proof may be found in [5] or [6].
It is now clear that the theory of realizations generalizes the theory of
elementary divisors: if A is some given matrix, then the set of all minimal
realizations of the sequence {Yk} = {Ak} is identical with the similarity
class of A, since all triples of the form (F, T, T-l), with T-IFT = A, are
minimal realizations. So the following is a well-defined problem: Given
{Yk} possessing a finite-dimensional realization, determine the similarity
invariants of F belonging to some minimal realization. A rather detailed
examination of this problem built around the classical machinery of invariant
factors and elementary divisors is given in [7], to which the reader is
referred also for additional motivation and background material.
In complete generality, that is, in terms of exhibiting efficient “functors”
to linear algebra, the solution of the problem is definitely not known at
present.? Let us review briefly what is known.
The simplest invariant of F (minimal) is given by the following result,
which is new, turns out to be quite simple, and seems to be fundamental:
Let S;denote the blockwise NXNgeneralized Hankel matrix
[Y#J Yl . . . YN-1 -
Yl Yz . . . YN
sy=. .
[&-1 Y, . . . y,-11
induced by the matrix sequence Y. Then dim F = rank ST for N suflciently
large. In other words, a finite-dimensional realization exists if and only if
the rank of S; is eventually constant, and then
n(Y) = rank SG = fl deg yi (vi+1 1 yip i = 1, . . ., q-l),
where yi are the invariant factors of the square matrix F belonging to any
t The computational experience of numerically determining minimal realizations has
been summarized in [8], pp. 373-405. Four methods have been compared there: (i) classical
elementary divisor theory (see [71); (ii) partial fraction expansions and rank computations
([4], Section 8); (iii) direct application of elementary linear algebra ([41 Sections
7 and 8); (iv) Ho’s algorithm via Hankel matrices 161.
minimal realization of Y. In particular, rank S; = n(Y) for all r =- n(Y) or
even r * deg ~1.
Referring to the module language mentioned at the end of 0 2, we can
rephrase the preceding statements also as follows: The module induced by
any F belonging to a minimal realization of Y is of dimension n(Y); this
module is the direct sum of precisely q cyclic pieces, each with annihilating
polynomial pi. In short, the (numerical) sequence Y may be used to determine
module invariants (n, q, the YJ with exactly the same ease (or difficulty)
as it can be used to determine similarity invariants for F.
It is clear that q, deg ~1, and even the vi could be determined by a combinatorial
procedure examining the linear dependences of subsets of the
matrix 5;. The detailed prescriptions are easily inferred from [4-81. It is,
however, not yet clear if a “functorial” procedure can be obtained for
this purpose. The clarification of this problem, in view of the situation
sketched above, is clearly one of the outstanding present research problems
in linear algebra.
This problem is closely related also to other unsolved elementary problems
of a linear-algebraic type. Let us mention the following interesting
CONJECTURE. (“Parametrization of minimal realizations.“) Let (F, G, ZZ)
be a minimal realization of its own sequence {Yk = HFkG, k = 0, 1, . . .}.
Let X be the corresponding K[z]-module, with q cyclic pieces. Let ~1, . . . , ye
be the invariant factors of X, with vi+1 ] yi and deg vi = n!. (Thus y1 = minimal
polynomial of X and nl + . . . + n, = dim X.) Finally, let m = p = q
and rank G = q.
Then: For fixed q and fixed (nl, . . . , n4) the set of all such triples plus
a set of measure zero (corresponding to triples which are not minimal)
is a linear space over K whose dimension is precisely equal to
5 min {ni, ?Zi,1 -ni}+ 2 (2j-1)nj (no = -).
i=l j=l
The first term in the above expression represents the minimal number
of parameters necessary to specify all the invariant factors (given their
degrees).
The second term is the dimension of the linear space of transformations
leaving the rational canonical form of F invariant. (This number was
first determined by Frobenius.) It can be shown that the dimension of
the linear transformations leaving F invariant is precisely the same as the
number of parameters in G which can be fixed over the whole class: For
instance, if q = 1 or if nl = . . . = nq all elements of G can be fixed.
This is closely related to canonical forms of completely controllable pairs
(F, G). See [7].