Talk:Circulant matrix

message: I'm not an expert, but I think the article is wrong in its description of diagonalizing a circulant matrix.  The article says that 
: C = F_n \operatorname{diag}(F_n c) F_n^{-1}, 
where
: F_n = (f_{jk}) \text{ with } f_{jk} = \mathrm{e}^{-2jk\pi i/n}.
While this gives the correct matrix of eigenvalues \operatorname{diag}(F_n c), I think the eigenvector matrices F_n should have a factor of 1/\sqrt{N}.  See, for example, http://ee.stanford.edu/~gray/toeplitz.pdf section 3.1 on the eigenvalues and eigenvectors of circulant matrices.  Also, the page on discrete Fourier transform matrix says that the factor of 1/\sqrt{N} is needed for unitarity.
72.130.185.73 (talk) 02:40, 8 April 2010 (UTC)
replies: text: Agreed.  I will fix this.  Oli Filth(talk|contribs) 07:07, 8 April 2010 (UTC)
thread_title:  Mistake in Diagonalizing Circulant Matrix? 

message: I'm looking to determine eigenvectors/eigenvalues of the matrix M=DC
where C is a circulant matrix and where D is a diagonal matrix whose i-th entry is z^{2i} where z is the n-th root of unity... Does anybody know how to do this?
thanks  —Preceding unsigned comment added by Asympt (talk • contribs) 02:32, 28 November 2009 (UTC)
thread_title:  Question about a generalization 

message: Is there a particular reason that the determinant of a circulant matrix is interesting (moreso than its eigenvalues)? I ask because listing the eigenvalues (which are the individual entries in the determinant product provided) would provide both pieces of information (since the determinant is always the product of the eigenvalues). However, only the determinant is provided now which says nothing about the individual eigenvalues. JokeySmurf (talk) 13:43, 28 July 2010 (UTC)
replies: text: The comparison of Determinant vs. Eigenvalues is irrelevant. Information of both quantities is valuable and should be present in the article. If you have something to add about eigenvalues to the article, please do. Maxal (talk) 14:13, 28 July 2010 (UTC)
thread_title: Determinant vs. Eigenvalues

message: This is with respect to a conflict I see in the eigenvalues with the definition stated here. From my understanding the eigenvalues are obtained as λ = c0 + c1*ωj + c2*ωj2 and so on where c0, c1, c2 and so on are the elements of the first row. But, this page gives the eigenvalues in terms of the first column which give us the eigenvalues but in an incorrect order. Please verify or if I am making a mistake somewhere, please clarify. 
http://mathworld.wolfram.com/CirculantDeterminant.html
The above link is for determinants but as can be seen, the eigenvalues are w.r.t first row.
Thank you.  —Preceding unsigned comment added by 69.143.35.92 (talk) 04:05, 19 November 2010 (UTC)
replies: text: The order of eigenvalues is arbitrary, so long as it matches the ordering of the eigenvectors. Plastikspork ―Œ(talk) 05:12, 19 November 2010 (UTC)

text: Thanks, fixed. Asympt (talk) 01:38, 21 November 2010 (UTC)
thread_title:  Definition clarification 

message: I mean, Weisstein does not appear anywhere in the body of the article... so how come there is an external link to him? Where it comes from?? It's not that i mind linking to him, but this should not be done automatically?? Asympt (talk) 02:06, 21 November 2010 (UTC)
replies: text: I removed the Weisstein page link. It was a stub that said little on the topic. 67.198.37.16 (talk) 23:52, 14 January 2024 (UTC)
thread_title:  Why is there a link to Weisstein page in external links 

message: Is that formula supposed to say something like "the real part of $\omega_v^k$"? Otherwise the eigenvectors will be complex, which doesn't make sense in the case of a symmetric circulant matrix. As an example, the matrix [[0,1,2,1],[1,0,1,2],[2,1,0,1],[1,2,1,0]] has eigenvector [1,0,-1,0]. The formula seems to predict in this case an eigenvector of (1, i, -1, -i).  —Preceding unsigned comment added by 135.207.103.61 (talk) 19:31, 23 February 2011 (UTC)
thread_title:  Mistake in Eigenvector formulation 

message: I added the term "anti-circulant matrix" and a redirect page for it to here.  I hope someone with more knowledge will add a section on them and how they are related to circulants, in particular the eigenvalues and characteristic polynomials.  And with references (which I was unable to find). Zaslav (talk) 02:53, 11 April 2020 (UTC)
thread_title:  Anti-circulant matrices 

message: I am surprised that the section on eigenvalues and eigenvectors does not state the characteristic polynomial.  Nowhere does the article discuss the form of the characteristic polynomial.  This should be fixed.  Covid-19 is blocking my access to a relevant textbook.   Zaslav (talk) 05:12, 17 April 2020 (UTC)
thread_title:  Characteristic polynomial 

message: I can find two definitions of block circulant matrices in the literature, and they are not compatible. Firstly, there is the one currently given: a block-circulant matrix is a matrix made up of square blocks, and the *square blocks* circulate according to the general pattern. But there's also the definition used in at least some cryptographic applications, such as the BIKE proposal, which states that "A binary circulant matrix is a square matrix where each row is the rotation of one element to the right of the preceding row. It is completely defined by its first row. A block-circulant matrix is formed of circulant square blocks of identical size". In other words, in their definition, all the blocks can be distinct and unrelated, but within the blocks, each block is a circulant matrix. These are not the same (though they feel in some sense dual to one another?) and I have been unable to find a definitive source either way.
I propose we remove the sentence about block circulants from this article until this can be resolved. LinesInk (talk) 15:47, 22 July 2022 (UTC)
thread_title:  Block-circulant matrices