linalg::frobeniusForm --
Frobenius form of a matrix
Introductionlinalg::frobeniusForm(A) returns the
Frobenius form of the matrix A, also called the Rational
Canonical form of A.
Call(s)linalg::frobeniusForm(A <, All>)
ParametersA |
- | a square matrix of a domain of category Cat::Matrix |
OptionsAll |
- | returns the list [R, P] with the
Frobenius form R of A and a transformation
matrix P such that A = P * R * P^(-1). |
Returnsa matrix of the same domain type as A, or the list
[R, P] when the option All is
given.
Related
Functionslinalg::jordanForm, linalg::hermiteForm,
linalg::smithForm,
linalg::minpoly
Detailslinalg::frobeniusForm(A, All) computes the Frobenius form R of
A and a transformation matrix P such that
A = P * R * 1/P.linalg::frobeniusForm is unique (see below).A must be a field, i.e., a
domain of category Cat::Field.
Example
1The Frobenius form of the following matrix over C:
>> A := Dom::Matrix(Dom::Complex)(
[[1, 2, 3], [4, 5, 6], [7, 8, 9]]
)
+- -+
| 1, 2, 3 |
| |
| 4, 5, 6 |
| |
| 7, 8, 9 |
+- -+
is the matrix:
>> R := linalg::frobeniusForm(A)
+- -+
| 0, 0, 0 |
| |
| 1, 0, 18 |
| |
| 0, 1, 15 |
+- -+
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]
+- -+
| 1, 1, 30 |
| |
| 0, 4, 66 |
| |
| 0, 7, 102 |
+- -+
We check the result:
>> P * R * P^(-1)
+- -+
| 1, 2, 3 |
| |
| 4, 5, 6 |
| |
| 7, 8, 9 |
+- -+
Background
+---------+
|R[1] 0|
| . |
R = | . | ,
| . |
| 0 R[r]|
+---------+
where R[1],...,R[r] are known as companion matrices and have
the form:
+-----------------+
|0 -a[0] |
|1 . |
R[i] = | . . |, i = 1..r.
| 1 0 -a[n[i]-1]|
+-----------------+
In the last column of the companion matrix R[i], you see the coefficients of its minimal polynomial in ascending order, i.e., the polynomial m[i]:= X^n[i] + a[n[i]-1]*X^(n[i]-1) + ... + a[1]*X + a[0] is the minimal polynomial of the matrix R[i].
For these polynomials the following holds: m[i] divides m[i+1] for i=1..r-1, and m[r] is the minimal polynomial of A.
The Frobenius form defined in this way is unique.