Frobenius form of a matrix
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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 transformation matrix P such
that P R P-
The Frobenius form as computed by
unique (see below).
The component ring of
A must be a field,
i.e., a domain of category
The Frobenius form of the following matrix over ℂ:
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
linalg::frobeniusForm with option
P := linalg::frobeniusForm(A, All)
We check the result:
P * R * P^(-1)
A square matrix of a domain of category
Returns the list
Matrix of the same domain type as
A, or the
[R, P] when the option
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 - 1 Xni - 1 + … + a1 X + 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