In linear algebra, the Frobenius normal form or rational canonical form of a square matrix A is a matrix canonical form 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.

Motivating example

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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:

$A=\backslash begin\{pmatrix\}$

-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

$A\_1\; =\; \backslash begin\{pmatrix\}$

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:

$M\; =\; A\_1\; \backslash oplus\; A\_2\; \backslash oplus\; A\_3\; =\; \backslash begin\{pmatrix\}$

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

$P\; =\; \backslash begin\{pmatrix\}$

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}.

General case and theory

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Fix a base field F and consider a finite-dimensional vector space V over F. Given a polynomial p(x) ∈ F^*, 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 Fhref="http://articles.gourt.com/en/module (mathematics)">module with the action of x given by A and extending by linearity) satisfies the F[x-module isomorphism

V ≅ F⊕ … ⊕ F[x/(a_n(x))

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

a₁(x) | … | a_n(x)

where "a | b" is notation for "a divides b".

Sketch of Proof: Apply the classification of finitely generated modules over principal ideal domains to V, viewing it as an FNote that any free Ffree 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" target="_blank" >^* 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 a_{i(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 a_i(x) is a polynomial in F^*, we may associate a corresponding block matrix C_i which is the companion matrix to a_i(x). In particular, each such C_i 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.

Reference

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• ^* David S. Dummit and Richard M. Foote. Abstract Algebra. 2nd Edition, John Wiley & Sons. pp. 442, 446, 452-458. ISBN 0471368571.

External links

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• Rational Canonical Form (Mathworld)

Algorithms

• An O(n³) Algorithm for Frobenius Normal Form
• An Algorithm for the Frobenius Normal Form (pdf)
• A rational canonical form Algorithm (pdf)

Linear algebra | Matrix normal forms

Frobenius-Normalform | צורה רציונלית