and
with elements in
polynomial domain F[λ] are called equivalent provided there exist non-singular matrices
and
such that
Website owner: James Miller
Elementary operations on lambda matrices, Equivalence, Smith normal form, invariant factors, elementary divisors
Elementary operations on λ-matrices. The elementary operations on λ-matrices are the same as those for regular matrices, except in Operation 3 the word “constant” is replaced by “polynomial”. An elementary operation on a λ-matrix over polynomial domain F(λ) consists of one of the following operations:
1. Interchanging any two rows or any two columns.
2. Multiplying any row or column by a non-zero constant.
3. Adding to any row the product of any arbitrary polynomial p(x) of F(λ) times another row or adding to any column the product of any arbitrary polynomial p(x) of F(λ) times another column.
We denote the different operations as follows:
Hij – interchange of the i-th and j-th rows
Kij – interchange of the i-th and j-th columns
Hi(k) – multiplication of the i-th row by the non-zero constant k
Ki(k)– multiplication of the i-th column by the non-zero constant k
Hij(p(λ)) – addition to the i-th row the product of p(λ) times the j-th row
Kij(p(λ)) – addition to the i-th column the product of p(λ) times the j-th column
As with regular matrices, corresponding to each operation is an elementary matrix, obtained by performing the elementary operation on the identity matrix In. And it works in the same way as with regular matrices – multiplication by the elementary matrix effects the same transformation as does the operation. We will denote the elementary matrix by the same symbol as we use for the operation.
The following theorems parallel those for regular matrices:
1] Every elementary matrix has an inverse which in turn is an elementary matrix.
2] Every non-singular λ-matrix can be expressed as a product of elementary matrices.
3] The rank of a λ-matrix is invariant under elementary operations.
Equivalence of λ-matrices. Two n-square λ-matrices
and
with elements in
polynomial domain F[λ] are called equivalent provided there exist non-singular matrices
and
such that
Two mxn λ-matrices are equivalent if and only if they have the same rank.
Theorem. Let
and
be equivalent matrices of rank r. Then the greatest common
divisor of all s-square minors of
, s ≤ r, is also the greatest common divisor of all s-square
minors of
.
Smith Normal Form. The Smith Normal Form is a canonical matrix of the form
(1)
=
where
1) Each
is a monic polynomial in λ (i.e. each
is a polynomial in which
the coefficient of the highest power of λ is unity)
2) Each
divides
(i = 1, 2, ....., r-1)
Theorem. Every λ-matrix
of rank r can be reduced to the Smith Normal Form by
elementary row and column transformations.
That Smith normal matrix
to which a given matrix A(λ) reduces by elementary row and
column transformations is uniquely determined by A(λ). Thus the Smith normal matrices form
a canonical set for equivalence over F(λ).
Invariant factors of a Lambda-matrix. The monic polynomials
in the diagonal of the Smith normal form of a Lambda-matrix are
called the invariant factors of the Lambda-matrix.
Trivial invariant factors of a Lambda-matrix. If an invariant factor of a Lambda-matrix is “1" it is called a “trivial invariant factor”. The first several diagonal elements in the Smith normal form of a Lambda-matrix may be “1" as in
If
= 1, k ≤ r, then
Theorem. Two n-square λ-matrices over F[λ] are equivalent over F[λ] if and only if they have the same invariant factors.
Elementary divisors of a Lambda-matrix. Each invariant factor of an n-square
λ-matrix over f[λ] can be expressed as a product of factors which are irreducible with respect to some number field (i.e. the field of rationals, reals, etc.). Since these irreducible factors may occur in multiples (i.e. powers of an irreducible factor may occur) we can say that an invariant factor f(λ) can be expressed in terms of its irreducible factors as
where
are distinct, monic, irreducible polynomials of F[λ].. Each
quantity
is called an elementary divisor of the corresponding Lambda-matrix. Thus an elementary divisor of a Lambda-matrix is an irreducible factor as raised to some power (the power being the number of times it occurs as a factor of the corresponding invariant factor).
Note. Over the complex field each invariant factor
can be expressed as a product of the
type
where
are distinct. Each factor
is an elementary divisor of the
Lambda-matrix.
Example. Suppose a 10-square λ-matrix
over the rational field has the Smith normal
form
The rank is 5.
The invariant factors are
The elementary divisors are:
Note that the elementary divisors are not necessarily distinct; In the listing each elementary divisor appears as often as it appears in the invariant factors.
Theorem. Two n-square λ-matrices over F[λ] are equivalent over F[λ] if and only if they have the same rank and the same elementary divisors.
References
Ayres. Matrix Theory. Chap. 24
Bocher. Introduction to Higher Algebra. Chap. XX