Elementary divisors
of a matrix over a polynomial ring
Powers of the monic irreducible polynomials over the field into which the invariant factors of
split. Two
-matrices over
having the same rank are equivalent (that is, can be obtained from one another by means of elementary operations) if and only if they have the same system of elementary divisors.
The elementary divisors of an -matrix
over
are, by definition, those of its characteristic matrix
. They can be obtained in the following manner. Let
be the greatest common divisor of the minors of order
of the matrix
,
, and let
. Then the invariant factors of
are
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The factors different from
are in
. Each of them can be represented in the form
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where the are monic irreducible polynomials over
,
, and
when
. All the polynomials
thus obtained form the system of elementary divisors of
. Two square matrices over a field are similar if and only if they have the same system of elementary divisors. The product of all elementary divisors of a matrix over a field is its characteristic polynomial, and their least common multiple is its minimum polynomial. Any collection of polynomials of the form
, where
is a monic irreducible polynomial over
, is the system of elementary divisors for one and only one class of similar matrices over
of order
, where
is the degree of the product of the
.
If is a splitting field of the characteristic polynomial of
, then the elementary divisors have the form
. Their number is then the same as the number of Jordan cells in the Jordan form of
, and the elementary divisor
corresponds to a Jordan cell
of order
(see Jordan matrix). A square matrix over a field
is similar to a diagonal matrix if and only if each elementary divisor of it has the form
, where
.
Comments
References
[a1] | N. Jacobson, "Lectures in abstract algebra" , II. Linear algebra , v. Nostrand (1953) |
[a2] | N. Jacobson, "Basic algebra" , I , Freeman (1974) |
Elementary divisors. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elementary_divisors&oldid=46800