Frobenius normal form

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In linear algebra, the Frobenius normal form, Turner binormal projective form or rational canonical form of a square matrix A is a canonical form for matrices that reflects the structure of the minimal polynomial of A and provides a means of detecting whether another matrix B is similar to A without extending the base field F.

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[edit] Motivating example

The Frobenius normal form M of a matrix A is generally calculated by means of a similarity transformation. What this implies is that there is an invertible matrix P so P−1AP = M. Since every matrix has a characteristic polynomial, then every matrix can be transformed by a similarity transformation to its Frobenius normal form.

For example, consider:

A=\begin{pmatrix} -2 & -1 & -2 & -1 & 1 & 0 \\ -2 & -1 & -2 & -1 & 1 & 1 \\ 2 & 1 & 2 & 1 & 0 & 0 \\ 2 & 1 & 0 & 1 & -3 & -1\\ -2 & 0 & -2 & 0 & 0 & 0 \\ 2 & -2 & 0 & 0 & 0 & 0 \end{pmatrix}.

The characteristic polynomial of A is x6+6x4+12x2+8 = (x2 + 2)3 = p(x). The minimal polynomial of this matrix is x2 + 2.

We will then have one block in the Frobenius normal form as

A_1 = \begin{pmatrix} 0 & -2 \\ 1 & 0 \end{pmatrix}.

The characteristic polynomial of A1 is indeed x2 +2.

Since p factors into solely terms of the form x2 + 2, we expect the other two blocks comprising the normal form of the matrix to be identical to A1. So, we can simply write down the Frobenius normal form:

M = A_1 \oplus A_2 \oplus A_3 = \begin{pmatrix} 0 & -2 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -2 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -2 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}.

The characteristic and minimal polynomials of M are the same as those of A, which we would expect, since M can be obtained via a similarity transformation P−1AP=M, and determinants are similarity invariant. For this matrix A, P is

P = \begin{pmatrix} 3/2 & -2 & 1/2 & 0 & 3/2 & 0 \\ 0 & -2 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & -2 & 0 & 0 \\ 1 & -3 & 1 & -1 & 1 & -1 \\ 0 & 3 & 0 & 1 & 0 & 3 \end{pmatrix}.

[edit] General case and theory

Fix a base field F and consider a finite-dimensional vector space V over F. Given a polynomial p(x) ∈ F[x], there is associated to it a companion matrix C whose characteristic polynomial is p(x). If the field F contains the eigenvalues of C (i.e., the roots of p(x)), then p(x) factors into linear factors over F. In this case, one can show that there is a Jordan normal form for C which in some sense comes as close to diagonalizing C as possible. When the base field F does not contain the eigenvalues of C, it is not possible to construct the Jordan normal form and another canonical form must be used. This is where the Frobenius normal form aids us. We recall the following

Theorem: Let V be a finite-dimensional vector space over a field F, and A a square matrix over F. Then V (viewed as an F[x]-module with the action of x given by A and extending by linearity) satisfies the F[x]-module isomorphism

V ≅ F[x]/(a1(x)) ⊕ … ⊕ F[x]/(an(x))

where the ai(x) ∈ F[x] may be taken to be non-units, unique as monic polynomials, and can be arranged to satisfy the relation

a1(x) | … | an(x)

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[x]-module. Note that any free F[x]-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 [DF] for details.

Given an arbitrary square matrix, the elementary divisors used in the construction of the Jordan normal form do not exist over F[x], so the invariant factors ai(x) as given above must be used instead. These correspond to factors of the minimal polynomial m(x) = an(x), which (by the Cayley-Hamilton theorem) itself divides the characteristic polynomial p(x) and in fact has the same roots as p(x), not counting multiplicities. Note in particular that the Theorem asserts that the invariant factors have coefficients in F.

As each invariant factor ai(x) is a polynomial in F[x], we may associate a corresponding block matrix Ci which is the companion matrix to ai(x). 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.

[edit] References

  • [DF] David S. Dummit and Richard M. Foote. Abstract Algebra. 2nd Edition, John Wiley & Sons. pp. 442, 446, 452-458. ISBN 0-471-36857-1.

[edit] External links

[edit] Algorithms