Frobenius normal form

In linear algebra, the Frobenius normal form (also known as the rational canonical form) of a matrix is a normal form, that consists of the direct sum of companion matrices which in turn correspond to certain monic polynomials. One of these may be the minimal polynomial of the matrix, or the minimal polynomial may be the same as the characteristic polynomial for this matrix, which means that the Frobenius normal form of the matrix will be the companion matrix of the characteristic polynomial.

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

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 M_is are the companion matrices of what are known as the invariant polynomials of M. If M_i is the companion matrix of a polynomial f_i(x), and the characteristic polynomial of M is p(x), then

$p(x)=f_{1}(x)f_{2}(x)\cdots f_{t}(x).$

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

External links

• 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)