ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 31 Dec 2018 15:46:24 -0600Finding the kernel of a matrix in a non-integral domainhttps://ask.sagemath.org/question/44815/finding-the-kernel-of-a-matrix-in-a-non-integral-domain/I have been trying to find the kernel of the matrix in a quotient, for example.
If we have the following quotient ring in sage:
R.<t> = PolynomialRing(GF(3),'t')
I = R.ideal([t^3])
S = R.quotient_ring(I);
and if I try to find the kernel of the matrix:
E = Matrix(S, ([[0+a*t+b*t^2, 1+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2, 1+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2,0+a*t+b*t^2]]))
E.kernel()
It gives me the following error: NotImplementedError.
I guess this is because F3[x]/(x^3) is not an integral domain but I would like a way around it.
Thanks in advance.Sun, 30 Dec 2018 12:11:11 -0600https://ask.sagemath.org/question/44815/finding-the-kernel-of-a-matrix-in-a-non-integral-domain/Comment by tmonteil for <p>I have been trying to find the kernel of the matrix in a quotient, for example.
If we have the following quotient ring in sage: </p>
<pre><code>R.<t> = PolynomialRing(GF(3),'t')
I = R.ideal([t^3])
S = R.quotient_ring(I);
</code></pre>
<p>and if I try to find the kernel of the matrix:</p>
<pre><code>E = Matrix(S, ([[0+a*t+b*t^2, 1+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2, 1+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2,0+a*t+b*t^2]]))
E.kernel()
</code></pre>
<p>It gives me the following error: NotImplementedError. </p>
<p>I guess this is because F3[x]/(x^3) is not an integral domain but I would like a way around it. </p>
<p>Thanks in advance.</p>
https://ask.sagemath.org/question/44815/finding-the-kernel-of-a-matrix-in-a-non-integral-domain/?comment=44827#post-id-44827Did you fix them or are them indeterminates ("variables") ?Mon, 31 Dec 2018 15:46:24 -0600https://ask.sagemath.org/question/44815/finding-the-kernel-of-a-matrix-in-a-non-integral-domain/?comment=44827#post-id-44827Comment by abel for <p>I have been trying to find the kernel of the matrix in a quotient, for example.
If we have the following quotient ring in sage: </p>
<pre><code>R.<t> = PolynomialRing(GF(3),'t')
I = R.ideal([t^3])
S = R.quotient_ring(I);
</code></pre>
<p>and if I try to find the kernel of the matrix:</p>
<pre><code>E = Matrix(S, ([[0+a*t+b*t^2, 1+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2, 1+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2,0+a*t+b*t^2]]))
E.kernel()
</code></pre>
<p>It gives me the following error: NotImplementedError. </p>
<p>I guess this is because F3[x]/(x^3) is not an integral domain but I would like a way around it. </p>
<p>Thanks in advance.</p>
https://ask.sagemath.org/question/44815/finding-the-kernel-of-a-matrix-in-a-non-integral-domain/?comment=44820#post-id-44820a and b are in F3. yes, i should have mentioned.Sun, 30 Dec 2018 22:58:38 -0600https://ask.sagemath.org/question/44815/finding-the-kernel-of-a-matrix-in-a-non-integral-domain/?comment=44820#post-id-44820Comment by tmonteil for <p>I have been trying to find the kernel of the matrix in a quotient, for example.
If we have the following quotient ring in sage: </p>
<pre><code>R.<t> = PolynomialRing(GF(3),'t')
I = R.ideal([t^3])
S = R.quotient_ring(I);
</code></pre>
<p>and if I try to find the kernel of the matrix:</p>
<pre><code>E = Matrix(S, ([[0+a*t+b*t^2, 1+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2, 1+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2,0+a*t+b*t^2]]))
E.kernel()
</code></pre>
<p>It gives me the following error: NotImplementedError. </p>
<p>I guess this is because F3[x]/(x^3) is not an integral domain but I would like a way around it. </p>
<p>Thanks in advance.</p>
https://ask.sagemath.org/question/44815/finding-the-kernel-of-a-matrix-in-a-non-integral-domain/?comment=44818#post-id-44818How do you define `a` and `b` ?Sun, 30 Dec 2018 15:21:18 -0600https://ask.sagemath.org/question/44815/finding-the-kernel-of-a-matrix-in-a-non-integral-domain/?comment=44818#post-id-44818Answer by rburing for <p>I have been trying to find the kernel of the matrix in a quotient, for example.
If we have the following quotient ring in sage: </p>
<pre><code>R.<t> = PolynomialRing(GF(3),'t')
I = R.ideal([t^3])
S = R.quotient_ring(I);
</code></pre>
<p>and if I try to find the kernel of the matrix:</p>
<pre><code>E = Matrix(S, ([[0+a*t+b*t^2, 1+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2, 1+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2,0+a*t+b*t^2]]))
E.kernel()
</code></pre>
<p>It gives me the following error: NotImplementedError. </p>
<p>I guess this is because F3[x]/(x^3) is not an integral domain but I would like a way around it. </p>
<p>Thanks in advance.</p>
https://ask.sagemath.org/question/44815/finding-the-kernel-of-a-matrix-in-a-non-integral-domain/?answer=44821#post-id-44821Since the ring is finite of order $3^3$ and the matrix is $4 \times 4$, one approach is to try every possible vector; there are only $(3^3)^4 = 531441$ possibilities.
A more refined approach is to introduce a vector with $3 \cdot 4 = 12$ undetermined coefficients, e.g. by making the base ring the quotient by $t^3$ of the univariate polynomial ring in $t$ over a polynomial ring in $12$ variables over $\mathbb{F}_3$.
Considering the matrix over this base ring, you can multiply with this vector with undetermined coefficients, take the components, and set the coefficients of powers of $t$ (of which the higher ones were automatically eliminated, due to the quotient by $t^3$) equal to zero; this is an ordinary $12 \times 12$ linear system over $\mathbb{F}_3$.Mon, 31 Dec 2018 00:11:05 -0600https://ask.sagemath.org/question/44815/finding-the-kernel-of-a-matrix-in-a-non-integral-domain/?answer=44821#post-id-44821