Similarity of matrices over a ring

106 Pages·1978·2.773 MB·English

by  Edith Mary McMahon

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MCMAHON* EDITH MARY
SIMILARITY OF MATRICES OVER A RING.
NORTHWESTERN UNIVERSITY, PH.D., 1973
University
Microfilms
International 300 n. zeeb roao, ann arbor, mi 48io6
NORTHWESTERN UNIVERSITY
SIMILARITY OF MATRICES
OVER A RING
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
for the degree
DOCTOR OF PHILOSOPHY
Field of Mathematics
By
Edith Mary McMahon
Evanston, Illinois
June, 1978
Table of Contents
Page
Introduction................. 1
Chapter 1: The Module Associated to a Matrix........... 11
1. Definition of the Module......................... 11
2. The Annihilator of M in R [ A ] ................ 12
3. Similarity of Matrices and Isomorphism
of Modules.......................................... 17
4. Properties of the Rings A and A ......... 19
Chapter 2: Lattices Over Orders ............... 25
1. Definitions and Basic Results .................. 25
2. The Jordan-Zassenhaus Condition..... ........... 27
3. Genera of Lattices................................. 33
Chapter 3: The Latimer-MacDuffee Correspondence . . . 40
1. The Classical Method.............................. 40
2. The Relationship of Modules and Ideals. . . . 47
Chapter 4: Block Triangular Form. ....................... 53
1. Transforming a Matrix into Block Triangular
Form................................................. 53
2. Similarity of Matrices in Block Triangular
Form.................................................. 63
Chapter 5: Determination of Similarity.................. 69
i
Page
1. Introductory Remarks........................... . 69
2. Form 1: £(X) has distinct irreducible
factors..................................................69
3. Form 2: All the roots of f(A) are the
same, f(A) = (A - a)n .............................. 71
4. Form 3: f(A) = g(A)(A - a) , and g(a) ^ 0 . 81
5. Form 4: f(A) = g(A)h(A) and h(A) has
degree 2 ...............................................85
6 . Summary.................................................. 93
Bibliography....................................................... 98
Vita................................................................101
i i
INTRODUCTION
The purpose of this dissertation is to study two
aspects of similarity of square matrices over a ring R .
We first discuss the number of similarity classes, shox^ing
when this number is finite. We then turn to the specific
problem of determining whether two square matrices, A and
A' , are similar over R . That is, we wish to determine
whether there exists an invertible matrix P over R such
that AP = PA' . We will first describe our methods and
approaches to these problems, and then make a comparison of
our work with results previously obtained by others.
The first two chapters are primarily concerned with
determining when the number of similarity classes is finite.
We begin by considering an arbitrary commutative ring, R ,
with identity. We associate an R [A]-module, Ma , with
each element A of Rn , the ring of n x n matrices with
elements from R . After proving that the annihilator of
Ma in R [A] is the null ideal of the matrix A and
therefore contains the characteristic polynomial, f(A) ,
of A , we consider M as a A = R [A]/(f(X))-module in the
obvious way. We note that, by its construction, MA is a
finitely generated A-module which is free as an R-module,
a fact which will be important in the second chapter.
2
A necessary condition for two matrices, A and B ,
of Rn to be similar is that they have the same charac­
teristic polynomial, f(A) . Thus, the associated modules,
MA and Mg respectively, are both A-modules. We then
prove, in Theorem 1.5, that two such matrices are similar
if and only if their associated modules are A-isomorphic.
The rest of the first chapter presents definitions and
results which are referred to in later chapters. We include
a discussion of how to determine the null ideal of matrix.
We also consider the ring A = Q[A]/(f(A)) , where Q is
the total quotient ring of R . We show that A is a
subring of A and that A is integral over R . Further,
the integral closure of R in A is shown to be the
integral closure of A in A . We give necessary and
sufficient conditions for A to be a separable Q-algebra.
The second chapter is concerned with the classification,
up to isomorphism, of the A-modules associated to matrices.
Throughout the chapter R is a Dedekind domain with
quotient field Q and A is a finite-dimensional Q-algebra.
The approach we take is to use the theory of lattices over
orders. An R-order S in A is a subring of A ,
containing the identity, such that S is a finitely
generated R-module which contains a Q-basis of A . An
S-lattice is an S-module which is finitely generated and
torsion-free as an R-module. We show that A is an R-
order in A and that MA is a A-lattice for any element
A of Rn .
Historically, R-orders were first studied in the
content of algebraic number theory. A 1 -order, for the
ring of rational integers, / , was defined to be a subring
O' , of the rings of the integers 0 of an algebraic
number field. Dedekind [7] studied the ideal class of O'
showing it to be finite and a multiple of the class number
of 0 .
The theory of lattices over orders has also been used
in representation theory. Roggenkamp and Huber-Dyson [25,
26] as well as Curtis and Reiner [6] stress this approach
to representation theory. If H is a finite group, the
group ring RH is an R-order in QH . Each RH-lattice wil
afford a set of R-equivalent R-representations of H .
This subject is developed in [6], A summary of the develop
ment for the special case when R = 1 is given in [29].
A survey of integral representation theory can be found in
[24]. In this case, we see a direct relationship between
matrices and integral representations. If f(A) is an
irreducible polynomial in 1 [X] and 0 is a root of
f(A) = 0 , then each matrix A such that f(A) = 0 define
an irreducible representation of ~J_ [0] . If P is an
invertible matrix, then P_1AP defines an equivalent
representation [22]. Thus, the number of inequivalent
irreducible representations of ~][ [0] is the same as the
number of similarity classes of matrix roots of f(A) = 0 .
From our work in Chapter 1, we know this to be the number
of isomorphism classes of ~jj [A]/(f(A)) lattices asso­
ciated to such matrices.
We will show that this number of isomorphism classes
of A-lattices is finite when R is a Dedekind domain whose
quotient field Q is a global field and A is a separable
Q-algebra. We do this by showing that the Jordan-Zassenhaus
condition, as stated in [23], holds for A-lattices, using
the fact that the unique maximal R-order in A is the
integral closure of R in A . Thus, we show that the
number of similarity classes of matrices with characteristic
polynomial f(A) is finite.
A different approach to the question of the number of
isomorphism classes of A-lattices is given by Jacobinski
[10,11]. He develops the classification of A-lattices into
genera and restricted genera. The number of classes is
shown to be finite and can be given in terms of the ideal
class number of a maximal R-order. We present these results
as they apply to the A-lattices associated to matrices.
We then use these results in an example showing when two
matrices are similar if and only if their associated A-
lattices are in the same genus.
We now turn to the problem of determining whether two
given matrices are similar. Throughout this portion of
the thesis, we assume that R is the ring of integers of
an algebraic number field with class number one, thus
insuring unique factorization. The method used in a parti­
cular situation depends on the form of the characteristic
5
polynomial f(A) . The factorization of f(A) in R[A]
will tell us the nature of the roots of f(A) = 0 and
whether A is separable. We note that when the irreducible
factors of f(A) are distinct, A is a separable Q-algebra
and we will have a finite number of similarity classes.
If the factors are not distinct, we may have an infinite
number of classes. A special case which illustrates this
is discussed in detail in Chapter 5.
If f(A) is irreducible over R or has distinct
irreducible factors with f(0) f 0 , we can associate an
ideal with each matrix. The matrices then are similar if
and only if the associated ideals are in the same class.
The theoretical basis for this correspondence is presented
in Chapter 3. The results given there are generalizations
of those of Latimer and MacDuffee. In [15] they establish
a one-to-one correspondence between similarity classes of
matrices in , which have f(A) as characteristic
polynomial, and classes of ideals in the ring of polynomials
~]L [C] , where C is the companion matrix of f(A) .
Taussky [30] specializes the correspondence to the case
\\rhere f(A) is irreducible. In this case, the ideal asso­
ciated to the matrix is specifically described. We present
all of these results for the more general case when R is
the ring of integers of an algebraic number field of class
number one. We also point out the connection between
ideals resulting from the Latimer-MacDuffee correspondence
and the modules defined in Chapter 1. We show that an