Editing Frobenius normal form (section)

== Motivation ==

When trying to find out whether two square matrices ''A'' and ''B'' are similar, one approach is to try, for each of them, to decompose the vector space as far as possible into a [[direct sum]] of stable subspaces, and compare the respective actions on these subspaces. For instance if both are [[diagonalizable]], then one can take the decomposition into [[eigenspace]]s (for which the action is as simple as it can get, namely by a scalar), and then similarity can be decided by comparing [[eigenvalue]]s and their multiplicities. While in practice this is often a quite insightful approach, there are various drawbacks this has as a general method. First, it requires finding all eigenvalues, say as [[root of a polynomial|roots]] of the [[characteristic polynomial]], but it may not be possible to give an explicit expression for them. Second, a complete set of eigenvalues might exist only in an extension of the field one is working over, and then one does not get a proof of similarity over the original field. Finally ''A'' and ''B'' might not be diagonalizable even over this larger field, in which case one must instead use a decomposition into [[generalized eigenspace]]s, and possibly into [[Jordan block]]s.

But obtaining such a fine decomposition is not necessary to just decide whether two matrices are similar. The rational canonical form is based on instead using a direct sum decomposition into stable subspaces that are as large as possible, while still allowing a very simple description of the action on each of them. These subspaces must be generated by a single nonzero vector ''v'' and all its images by repeated application of the [[linear operator]] associated to the matrix; such subspaces are called cyclic subspaces (by analogy with [[cyclic group|cyclic]] [[subgroup]]s) and they are clearly stable under the linear operator. A [[basis (linear algebra)|basis]] of such a subspace is obtained by taking ''v'' and its successive images as long as they are [[linearly independent]]. The matrix of the linear operator with respect to such a basis is the [[companion matrix]] of a [[monic polynomial]]; this polynomial (the [[minimal polynomial (linear algebra)|minimal polynomial]] of the operator [[restriction (mathematics)|restricted]] to the subspace, which notion is analogous to that of the [[order (group theory)|order]] of a cyclic subgroup) determines the action of the operator on the cyclic subspace up to [[isomorphism]], and is independent of the choice of the vector ''v'' generating the subspace.

A direct sum decomposition into cyclic subspaces always exists, and finding one does not require factoring polynomials. However it is possible that cyclic subspaces do allow a decomposition as direct sum of smaller cyclic subspaces (essentially by the [[Chinese remainder theorem]]). Therefore, just having for both matrices some decomposition of the space into cyclic subspaces, and knowing the corresponding minimal polynomials, is not in itself sufficient to decide their similarity. An additional condition is imposed to ensure that for similar matrices one gets decompositions into cyclic subspaces that exactly match: in the list of associated minimal polynomials each one must divide the next (and the constant polynomial 1 is forbidden to exclude [[zero vector space|trivial]] cyclic subspaces). The resulting list of polynomials are called the [[invariant factor]]s of (the ''K''[''X'']-[[module (mathematics)|module]] defined by) the matrix, and two matrices are similar if and only if they have identical lists of invariant factors. The rational canonical form of a matrix ''A'' is obtained by expressing it on a basis adapted to a decomposition into cyclic subspaces whose associated minimal polynomials are the invariant factors of ''A''; two matrices are similar if and only if they have the same rational canonical form.