ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 27 Feb 2024 16:01:31 +0100rational canonical formhttps://ask.sagemath.org/question/76030/rational-canonical-form/Please i want to compute the rational canonical form of a matrix also known as frobenius form in terms of the entries of the matrix
the matrix is a `2*2` matrix of the form `A=[[a*b+1,b],[a,1]`
Thank you in advanceFri, 16 Feb 2024 20:04:52 +0100https://ask.sagemath.org/question/76030/rational-canonical-form/Comment by miloulou for <p>Please i want to compute the rational canonical form of a matrix also known as frobenius form in terms of the entries of the matrix
the matrix is a <code>2*2</code> matrix of the form <code>A=[[a*b+1,b],[a,1]</code>
Thank you in advance</p>
https://ask.sagemath.org/question/76030/rational-canonical-form/?comment=76085#post-id-76085@emmanuel Charpentier
No it is not a homeworkMon, 19 Feb 2024 14:13:32 +0100https://ask.sagemath.org/question/76030/rational-canonical-form/?comment=76085#post-id-76085Comment by Emmanuel Charpentier for <p>Please i want to compute the rational canonical form of a matrix also known as frobenius form in terms of the entries of the matrix
the matrix is a <code>2*2</code> matrix of the form <code>A=[[a*b+1,b],[a,1]</code>
Thank you in advance</p>
https://ask.sagemath.org/question/76030/rational-canonical-form/?comment=76061#post-id-76061Homework ?Sun, 18 Feb 2024 16:10:45 +0100https://ask.sagemath.org/question/76030/rational-canonical-form/?comment=76061#post-id-76061Answer by stevediaz for <p>Please i want to compute the rational canonical form of a matrix also known as frobenius form in terms of the entries of the matrix
the matrix is a <code>2*2</code> matrix of the form <code>A=[[a*b+1,b],[a,1]</code>
Thank you in advance</p>
https://ask.sagemath.org/question/76030/rational-canonical-form/?answer=76230#post-id-76230To compute the rational canonical form (Frobenius form) of a matrix, you need to find the invariant factors. For a 2x2 matrix of the form
�
=
[
�
�
+
1
�
�
1
]
A=[
ab+1
a
b
1
], we can find its characteristic polynomial and minimal polynomial, and then use them to determine the rational canonical form.
Let's start by finding the characteristic polynomial. The characteristic polynomial is given by:
det
(
�
−
�
�
)
=
0
det(A−λI)=0
Where
�
I is the identity matrix.
det
[
(
�
�
+
1
)
−
�
�
�
1
−
�
]
=
0
det[
(ab+1)−λ
a
b
1−λ
]=0
Expanding this determinant gives:
(
(
�
�
+
1
)
−
�
)
(
1
−
�
)
−
�
�
=
0
((ab+1)−λ)(1−λ)−ab=0
Simplifying:
(
�
�
+
1
−
�
)
(
1
−
�
)
−
�
�
=
0
(ab+1−λ)(1−λ)−ab=0
(
�
�
+
1
−
�
−
�
+
�
2
)
−
�
�
=
0
(ab+1−λ−λ+λ
2
)−ab=0
(
1
−
�
+
�
2
)
=
0
(1−λ+λ
2
)=0
Now, you can solve this quadratic equation to find the eigenvalues
�
λ.
Once you have the eigenvalues, you can find the corresponding eigenvectors and use them to construct the Jordan chains. The rational canonical form will then consist of blocks corresponding to the Jordan chains.Mon, 26 Feb 2024 12:20:07 +0100https://ask.sagemath.org/question/76030/rational-canonical-form/?answer=76230#post-id-76230Comment by Max Alekseyev for <p>To compute the rational canonical form (Frobenius form) of a matrix, you need to find the invariant factors. For a 2x2 matrix of the form </p>
<h1>�</h1>
<p>[
�
�
+
1
�
�
1
]
A=[
ab+1
a
</p>
<p>b
1
], we can find its characteristic polynomial and minimal polynomial, and then use them to determine the rational canonical form.</p>
<p>Let's start by finding the characteristic polynomial. The characteristic polynomial is given by:</p>
<p>det
(
�
−
�
�</p>
<h1>)</h1>
<p>0
det(A−λI)=0</p>
<p>Where
�
I is the identity matrix.</p>
<p>det
[
(
�
�
+
1
)
−
�
�
�
1
−
�</p>
<h1>]</h1>
<p>0
det[
(ab+1)−λ
a
</p>
<p>b
1−λ
]=0</p>
<p>Expanding this determinant gives:</p>
<p>(
(
�
�
+
1
)
−
�
)
(
1
−
�
)
−
�</p>
<h1>�</h1>
<p>0
((ab+1)−λ)(1−λ)−ab=0</p>
<p>Simplifying:</p>
<p>(
�
�
+
1
−
�
)
(
1
−
�
)
−
�</p>
<h1>�</h1>
<p>0
(ab+1−λ)(1−λ)−ab=0</p>
<p>(
�
�
+
1
−
�
−
�
+
�
2
)
−
�</p>
<h1>�</h1>
<p>0
(ab+1−λ−λ+λ
2
)−ab=0</p>
<p>(
1
−
�
+
�
2</p>
<h1>)</h1>
<p>0
(1−λ+λ
2
)=0</p>
<p>Now, you can solve this quadratic equation to find the eigenvalues
�
λ.</p>
<p>Once you have the eigenvalues, you can find the corresponding eigenvectors and use them to construct the Jordan chains. The rational canonical form will then consist of blocks corresponding to the Jordan chains.</p>
https://ask.sagemath.org/question/76030/rational-canonical-form/?comment=76263#post-id-76263Your answer is unreadable with a lot of � characters. Also, where is Sage in your answer?Tue, 27 Feb 2024 16:01:31 +0100https://ask.sagemath.org/question/76030/rational-canonical-form/?comment=76263#post-id-76263Answer by Max Alekseyev for <p>Please i want to compute the rational canonical form of a matrix also known as frobenius form in terms of the entries of the matrix
the matrix is a <code>2*2</code> matrix of the form <code>A=[[a*b+1,b],[a,1]</code>
Thank you in advance</p>
https://ask.sagemath.org/question/76030/rational-canonical-form/?answer=76124#post-id-76124Like this:
sage: K = PolynomialRing(QQ,2,'x').fraction_field()
sage: x = K.gens()
sage: A = Matrix([[x[0]*x[1]+1,x[1]],[x[0],1]])
sage: A.rational_form()
[ 0 -1]
[ 1 x0*x1 + 2]Tue, 20 Feb 2024 19:08:20 +0100https://ask.sagemath.org/question/76030/rational-canonical-form/?answer=76124#post-id-76124