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Frobenius form of a matrix

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linalg::frobeniusForm(A, <All>)


linalg::frobeniusForm(A) returns the Frobenius form of the matrix A, also called the Rational Canonical form of A.

linalg::frobeniusForm(A, All) computes the Frobenius form R of A and a transformation matrix P such that PRP- 1.

The Frobenius form as computed by linalg::frobeniusForm is unique (see below).

The component ring of A must be a field, i.e., a domain of category Cat::Field.


Example 1

The Frobenius form of the following matrix over :

A := Dom::Matrix(Dom::Complex)( 
  [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

is the matrix:

R := linalg::frobeniusForm(A)

The transformation matrix P can be selected from the list [R, P], which is the result of linalg::frobeniusForm with option All:

P := linalg::frobeniusForm(A, All)[2]

We check the result:

P * R * P^(-1)



A square matrix of a domain of category Cat::Matrix



Returns the list [R, P] with the Frobenius form R of A and a transformation matrix P such that A = PRP- 1.

Return Values

Matrix of the same domain type as A, or the list [R, P] when the option All is given.


The Frobenius form of a square matrix A is the matrix


where R1, …, Rr are known as companion matrices and have the form:


In the last column of the companion matrix Ri, you see the coefficients of its minimal polynomial in ascending order, i.e., the polynomial mi := Xni + ani - 1Xni - 1 + … + a1X + a0 is the minimal polynomial of the matrix Ri.

For these polynomials the following holds: mi + 1 divides mi for i = 1, …, r - 1, and the product of all mi for i = 1, …, r gives a factorization of the characteristic polynomial of the matrix A. The Frobenius form defined in this way is unique.


Reference: P. Ozello: Calcul exact des formes de Jordan et de Frobenius d'une matrice, pp. 30–43. Thèse de l'Universite Scientifique Technologique et Medicale de Grenoble, 1987

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