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.Mon, 19 Oct 2015 04:54:11 +0200eigenvectors of complex matrixhttps://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/Hi!
I would like to find the complex eigenvectors of this matrix:
A=matrix(CDF,[[2-i,0,i],[0,1+i,0],[i,0,2-i]]).
I have used the command A.eigenvectors_right() and I get the following eigenvectors (rounded off):
(-0.70711+9.4136e-17i , 0 , 0.70711), (0,1,0), (0.70711 , 0 , 0.70711)
In my checklist I should get the vectors: t1*(-1,0,1), t2*(0,1,0), t3*(1,0,1), where the t-values are complex factors.
How do I compute this kind of result?
Sincerly SimonWed, 14 Oct 2015 09:11:18 +0200https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/Comment by vdelecroix for <p>Hi!</p>
<p>I would like to find the complex eigenvectors of this matrix:</p>
<pre><code>A=matrix(CDF,[[2-i,0,i],[0,1+i,0],[i,0,2-i]]).
</code></pre>
<p>I have used the command A.eigenvectors_right() and I get the following eigenvectors (rounded off):</p>
<pre><code>(-0.70711+9.4136e-17i , 0 , 0.70711), (0,1,0), (0.70711 , 0 , 0.70711)
</code></pre>
<p>In my checklist I should get the vectors: t1<em>(-1,0,1), t2</em>(0,1,0), t3*(1,0,1), where the t-values are complex factors.</p>
<p>How do I compute this kind of result?</p>
<p>Sincerly Simon</p>
https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?comment=29996#post-id-29996Do you know what is an eigenvector?Wed, 14 Oct 2015 14:11:41 +0200https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?comment=29996#post-id-29996Comment by ismon for <p>Hi!</p>
<p>I would like to find the complex eigenvectors of this matrix:</p>
<pre><code>A=matrix(CDF,[[2-i,0,i],[0,1+i,0],[i,0,2-i]]).
</code></pre>
<p>I have used the command A.eigenvectors_right() and I get the following eigenvectors (rounded off):</p>
<pre><code>(-0.70711+9.4136e-17i , 0 , 0.70711), (0,1,0), (0.70711 , 0 , 0.70711)
</code></pre>
<p>In my checklist I should get the vectors: t1<em>(-1,0,1), t2</em>(0,1,0), t3*(1,0,1), where the t-values are complex factors.</p>
<p>How do I compute this kind of result?</p>
<p>Sincerly Simon</p>
https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?comment=30002#post-id-30002It is the vector that corresponds to a given eigenvalue. It represents the vector by which translation to the same vectorspace can be done by a translation matrix or a eigenvalue. Eigenvectors can generate a basis for the vectorspace, with no higher dimension than the original vectorspace. I'm sorry if the explanation is not all correct, I'm used to discuss linear algebra in danish.
Anyways, how does your question answer mine?Wed, 14 Oct 2015 15:20:20 +0200https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?comment=30002#post-id-30002Comment by kcrisman for <p>Hi!</p>
<p>I would like to find the complex eigenvectors of this matrix:</p>
<pre><code>A=matrix(CDF,[[2-i,0,i],[0,1+i,0],[i,0,2-i]]).
</code></pre>
<p>I have used the command A.eigenvectors_right() and I get the following eigenvectors (rounded off):</p>
<pre><code>(-0.70711+9.4136e-17i , 0 , 0.70711), (0,1,0), (0.70711 , 0 , 0.70711)
</code></pre>
<p>In my checklist I should get the vectors: t1<em>(-1,0,1), t2</em>(0,1,0), t3*(1,0,1), where the t-values are complex factors.</p>
<p>How do I compute this kind of result?</p>
<p>Sincerly Simon</p>
https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?comment=30012#post-id-30012I think that @vdelecroix's point is that eigenvectors are not unique. Your checklist and the eigenvectors you got from Sage obtain exactly the same eigenspaces, so you are both right.Wed, 14 Oct 2015 17:28:43 +0200https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?comment=30012#post-id-30012Comment by ismon for <p>Hi!</p>
<p>I would like to find the complex eigenvectors of this matrix:</p>
<pre><code>A=matrix(CDF,[[2-i,0,i],[0,1+i,0],[i,0,2-i]]).
</code></pre>
<p>I have used the command A.eigenvectors_right() and I get the following eigenvectors (rounded off):</p>
<pre><code>(-0.70711+9.4136e-17i , 0 , 0.70711), (0,1,0), (0.70711 , 0 , 0.70711)
</code></pre>
<p>In my checklist I should get the vectors: t1<em>(-1,0,1), t2</em>(0,1,0), t3*(1,0,1), where the t-values are complex factors.</p>
<p>How do I compute this kind of result?</p>
<p>Sincerly Simon</p>
https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?comment=30018#post-id-30018The question is not whether my checklist answers or the output from sage is correct, I know they both are!
My question is how to formulate a command in sage so that the result will be of the same form as that in my checklist. It's not a question of getting a correct answer, but of how the output is displayed.Wed, 14 Oct 2015 23:16:47 +0200https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?comment=30018#post-id-30018Comment by vdelecroix for <p>Hi!</p>
<p>I would like to find the complex eigenvectors of this matrix:</p>
<pre><code>A=matrix(CDF,[[2-i,0,i],[0,1+i,0],[i,0,2-i]]).
</code></pre>
<p>I have used the command A.eigenvectors_right() and I get the following eigenvectors (rounded off):</p>
<pre><code>(-0.70711+9.4136e-17i , 0 , 0.70711), (0,1,0), (0.70711 , 0 , 0.70711)
</code></pre>
<p>In my checklist I should get the vectors: t1<em>(-1,0,1), t2</em>(0,1,0), t3*(1,0,1), where the t-values are complex factors.</p>
<p>How do I compute this kind of result?</p>
<p>Sincerly Simon</p>
https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?comment=30021#post-id-30021There are hundreds of different ways to write down eigenvectors. None of them is canonical. You should explain with more details what you want as output (not only an example).Thu, 15 Oct 2015 03:12:36 +0200https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?comment=30021#post-id-30021Comment by kcrisman for <p>Hi!</p>
<p>I would like to find the complex eigenvectors of this matrix:</p>
<pre><code>A=matrix(CDF,[[2-i,0,i],[0,1+i,0],[i,0,2-i]]).
</code></pre>
<p>I have used the command A.eigenvectors_right() and I get the following eigenvectors (rounded off):</p>
<pre><code>(-0.70711+9.4136e-17i , 0 , 0.70711), (0,1,0), (0.70711 , 0 , 0.70711)
</code></pre>
<p>In my checklist I should get the vectors: t1<em>(-1,0,1), t2</em>(0,1,0), t3*(1,0,1), where the t-values are complex factors.</p>
<p>How do I compute this kind of result?</p>
<p>Sincerly Simon</p>
https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?comment=30042#post-id-30042Agreed; most likely with a little string processing what you want is possible, but it's hard to tell exactly what you want from your example.Thu, 15 Oct 2015 17:00:28 +0200https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?comment=30042#post-id-30042Answer by tmonteil for <p>Hi!</p>
<p>I would like to find the complex eigenvectors of this matrix:</p>
<pre><code>A=matrix(CDF,[[2-i,0,i],[0,1+i,0],[i,0,2-i]]).
</code></pre>
<p>I have used the command A.eigenvectors_right() and I get the following eigenvectors (rounded off):</p>
<pre><code>(-0.70711+9.4136e-17i , 0 , 0.70711), (0,1,0), (0.70711 , 0 , 0.70711)
</code></pre>
<p>In my checklist I should get the vectors: t1<em>(-1,0,1), t2</em>(0,1,0), t3*(1,0,1), where the t-values are complex factors.</p>
<p>How do I compute this kind of result?</p>
<p>Sincerly Simon</p>
https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?answer=30109#post-id-30109What you get from Sage are vectors of *euclidean* norm one:
sage: [vector(CDF,i[1][0]).norm() for i in A.eigenvectors_right()]
[1.0, 1.0, 1.0]
If i try to interpret your checklist, it seems that you want to get eigenvectors of *infinite* norm 1. For this, you can normalize the given solution as follows:
sage: [vector(CDF,i[1][0])/vector(CDF,i[1][0]).norm(p = Infinity) for i in A.eigenvectors_right()]
[(1.0, 0.0, 0.9999999999999998),
(1.0, 0.0, -0.9999999999999998 - 2.2570079093579266e-16*I),
(0.0, 1.0, 0.0)]
Mon, 19 Oct 2015 04:54:11 +0200https://ask.sagemath.org/question/29989/eigenvectors-of-complex-matrix/?answer=30109#post-id-30109