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# Frobenius normal form

In linear algebra, the Frobenius normal form of a matrix is a normal form that reflects the structure of the minimal polynomial of a matrix.

The idea of the companion matrix of a polynomial demonstrates this relationship to some extent. If one has the companion matrix C of a polynomial p, the characteristic polynomial of C is p. But polynomials can of course be factored. Can the companion matrix then be simplified to reflect this fact? Yes it can, and this is where the Frobenius normal form aids us.

Given an arbitrary matrix, we can calculate block matrices which are companion matrices, each of which correspond to factors of the minimal polynomial. These blocks can be put together to create the Frobenius normal form.

The coefficients of the factors of the minimal polynomial become entries in the Frobenius normal form in a similar fashion to the companion matrix. Where the minimal polynomial is identical to the characteristic polynomial, the Frobenius normal form is the companion matrix of the characteristic polynomial. This fact has an important consequence. Since the matrix consists of elements from some field, and the characteristic polynomial consists of elements from the matrix, and the Frobenius normal form contains elements that are coefficients of the characteristic polynomial, then the Frobenius normal form is not constrained to the complexes as the Jordan normal form is, another normal form that is commonly used. This leads to the alternate name for this form, the rational canonical form (for example, if the matrix was over Q).

The Frobenius normal form F 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=F. Since every matrix has a characteristic polynomial, then every matrix can be transformed by a similarity transformation to its Frobenius normal form.

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## Motivating example

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 x2 +2 terms, 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:

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

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

$Q = \begin{pmatrix} 3/2 & -2 & 1/2 & 0 & 3/2 & 0 \\ 0 & -2 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & -2 & 0 & 0 \\ 1 & -3 & 1 & -1 & 1 & -1 \\ 0 & 3 & 0 & 1 & 0 & 3 \end{pmatrix}$

## General case

In general, if M is some n×n matrix, there is an invertible matrix P such that P−1MP=F where

$F = M_1 \oplus M_2 \oplus \cdots \oplus M_t$

where the Mis are the companion matrices of what are known as the invariant polynomials of M. If Mi is the companion matrix of a polynomial fi(x), and the characteristic polynomial of M is p(x), then

$p(x)=f_1(x)f_2(x)\cdots f_t(x).$

Each fi divides fi+1, if one writes the companion matrices Mi such that as i increases, so does the number of rows/columns of Mi. The polynomial ft then is the minimal polynomial of M.