Frobenius normal form


In linear algebra, the Frobenius normal form or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices obtained by conjugation by invertible matrices over F. The form reflects a minimal decomposition of the vector space into subspaces that are cyclic for A. Since only one normal form can be reached from a given matrix, a matrix B is similar to A if and only if it has the same rational canonical form as A. Since this form can be found without any operations that might change when extending the field F, notably without factoring polynomials, this shows that whether two matrices are similar does not change upon field extensions. The form is named after German mathematician Ferdinand Georg Frobenius.
Some authors use the term rational canonical form for a somewhat different form that is more properly called the primary rational canonical form. Instead of decomposing into a minimal number of cyclic subspaces, the primary form decomposes into a maximal number of cyclic subspaces. It is also defined over F, but has somewhat different properties: finding the form requires factorization of polynomials, and as a consequence the primary rational canonical form may change when the same matrix is considered over an extension field of F. This article mainly deals with the form that does not require factorization, and explicitly mentions "primary" when the form using factorization is meant.

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 eigenspaces, and then similarity can be decided by comparing eigenvalues 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 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 eigenspaces, and possibly into Jordan blocks.
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 and they are clearly stable under the linear operator. A 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 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. 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. The resulting list of polynomials are called the invariant factors of 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.

Example

Consider the following matrix A, over Q:
A has minimal polynomial, so that the dimension of a subspace generated by the repeated images of a single vector is at most 6. The characteristic polynomial is, which is a multiple of the minimal polynomial by a factor. There always exist vectors such that the cyclic subspace that they generate has the same minimal polynomial as the operator has on the whole space; indeed most vectors will have this property, and in this case the first standard basis vector does so: the vectors for are linearly independent and span a cyclic subspace with minimal polynomial. There exist complementary stable subspaces to this cyclic subspace, and the space generated by vectors and is an example. In fact one has, so the complementary subspace is a cyclic subspace generated by ; it has minimal polynomial. Since is the minimal polynomial of the whole space, it is clear that must divide , and we have found the invariant factors and of A. Then the rational canonical form of A is the block diagonal matrix with the corresponding companion matrices as diagonal blocks, namely
A basis on which this form is attained is formed by the vectors above, followed by for ; explicitly this means that for
one has

General case and theory

Fix a base field F and a finite-dimensional vector space V over F. Given a polynomial p ∈ F, there is associated to it a companion matrix C whose characteristic polynomial is p.
Theorem: Let V be a finite-dimensional vector space over a field F, and A a square matrix over F. Then V satisfies the F-module isomorphism
where the ai ∈ F may be taken to be non-units, unique as monic polynomials, and can be arranged to satisfy the relation
where "a | b" is notation for "a divides b".
Sketch of Proof: Apply the structure theorem for finitely generated modules over a principal ideal domain to V, viewing it as an F-module. Note that any free F-module is infinite-dimensional over F, so that the resulting direct sum decomposition has no free part since V is finite-dimensional. The uniqueness of the invariant factors requires a separate proof that they are determined up to units; then the monic condition ensures that they are uniquely determined. The proof of this latter part is omitted. See for details.
Given an arbitrary square matrix, the elementary divisors used in the construction of the Jordan normal form do not exist over F, so the invariant factors ai as given above must be used instead. These correspond to factors of the minimal polynomial m = an, which itself divides the characteristic polynomial p and in fact has the same roots as p, not counting multiplicities. Note in particular that the Theorem asserts that the invariant factors have coefficients in F.
As each invariant factor ai is a polynomial in F, we may associate a corresponding block matrix Ci which is the companion matrix to ai. In particular, each such Ci has its entries in the field F.
Taking the matrix direct sum of these blocks over all the invariant factors yields the rational canonical form of A. Where the minimal polynomial is identical to the characteristic polynomial, the Frobenius normal form is the companion matrix of the characteristic polynomial. As the rational canonical form is uniquely determined by the unique invariant factors associated to A, and these invariant factors are independent of basis, it follows that two square matrices A and B are similar if and only if they have the same rational canonical form.

A rational normal form generalizing the Jordan normal form

The Frobenius normal form does not reflect any form of factorization of the characteristic polynomial, even if it does exist over the ground field F. This implies that it is invariant when F is replaced by a different field. On the other hand, this makes the Frobenius normal form rather different from other normal forms that do depend on factoring the characteristic polynomial, notably the diagonal form or more generally the Jordan normal form. For instance, the Frobenius normal form of a diagonal matrix with distinct diagonal entries is just the companion matrix of its characteristic polynomial.
There is another way to define a normal form, that, like the Frobenius normal form, is always defined over the same field F as A, but that does reflect a possible factorization of the characteristic polynomial into irreducible factors over F, and which reduces to the Jordan normal form when this factorization only contains linear factors. This form is sometimes called the generalized Jordan normal form, or primary rational canonical form. It is based on the fact that the vector space can be canonically decomposed into a direct sum of stable subspaces corresponding to the distinct irreducible factors P of the characteristic polynomial, where the characteristic polynomial of each summand is a power of the corresponding P. These summands can be further decomposed, non-canonically, as a direct sum of cyclic F-modules, where the characteristic polynomial of each summand is still a power of P. The primary rational canonical form is a block diagonal matrix corresponding to such a decomposition into cyclic modules, with a particular form called generalized Jordan block in the diagonal blocks, corresponding to a particular choice of a basis for the cyclic modules. This generalized Jordan block is itself a block matrix of the form
where C is the companion matrix of the irreducible polynomial, and is a matrix whose sole nonzero entry is a 1 in the upper right hand corner. For the case of a linear irreducible factor, these blocks are reduced to single entries and and, one finds a Jordan block. In any generalized Jordan block, all entries immediately below the main diagonal are 1. A basis of the cyclic module giving rise to this form is obtained by choosing a generating vector , and taking as basis
where.

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