Talk:Perron–Frobenius theorem

message: Just wanted to leave a note that the hyperlink to meyer's chapter 8 is dead. 217.229.108.214 (talk) 07:36, 13 August 2014 (UTC)

message: *Please could you define (in this article or elsewhere) what you mean by a left and right eigenspaces and eigenvectors? Thanks. Lupin 04:21, 9 Mar 2005 (UTC)
replies: text: A recent edit said "positive" meant "= 0", so I reverted.  I'm being told on my discussion page that that was an error.  I see that there was one other small edit that I reverted along with it; I think it changed ">" to "≥" or something like that, and I'm not sure right now which is right. Michael Hardy 17:55, 26 September 2006 (UTC)
replies: text: OK, now I see that it did not say "= 0"; it said ">= 0".  If "greater than or equal to" was intended, then it should say "≥ 0". Michael Hardy 17:57, 26 September 2006 (UTC)

text: The theorem as stated applies only to positive matrices, that is, every entry is strictly > 0. Your matrix has zero entries, thus it won't apply. The theorem can be extended to any positive definite matrix, but your matrix is not positive definite (as its determinant is -1). Oops, missed that you were talking about the nonnegative section. Indeed you are right, something's funny there. -- dcclark (talk) 03:35, 13 February 2007 (UTC)
replies: text:  This page would suggest that a non-strict inequality applies in the non-negative case. Rawling4851 09:59, 15 February 2007 (UTC)
thread_title:  Definition 

message: There are a couple of things wrong here. The most obvious is the statement that "Every other eigenvector for smaller eigenvalues has negative components". The matrix (2 0 // 0 1) is a trivial counterexample. However it also overlooks the fact that smaller imaginary eigenvalues may exist whose eigenvectors will necessarily involve imaginary components. To remove any of these it's necessary to consider conjugate pairs of eigenvalues. 
My understanding is that Perron proved the theorem for positive matrices in 1907. In 1912 Frobenius extended it to non-negative irreducible matrices, and it's his version which is today known as the Perron-Frobenius theorem. In other words the P-F theorem applies solely to non-negative irreducible matrices and not non-negative matrices in general. However any non-negative matrix can be expressed in lower triangular block form where each diagonal block is either irreducible or zero, so the P-F theorem does unlock the door to results on general non-negative matrices. MRFS (talk) 14:47, 7 January 2009 (UTC)
replies: text: The changes regarding irreducibility in the section on non-negative matrices are not consistent.  The assumptions of the theorem now include irreducibility, but that is not reflected in the full strength of the results.  Two paragraphs down, irreducibility is treated as an additional assumption.  Someone with a strong knowledge of the theorem should clear this up. 132.163.229.170 (talk) 16:41, 21 April 2009 (UTC)
replies: text: Furthermore, in the first part irreducibility of the positive matrix is assumed. But in the "Perron-Frobenius theorem for non-negative matrices" section, there is written: "Note that a positive matrix is irreducible". If this is right, the additional assumption is confusing. Gernegross (talk) 16:29, 22 April 2009 (UTC)
replies: text: The rewrite seems helpful, but it needs to be sourced (Horn & Johnson?). The canonical form is due to Wielandt, I believe, and it is not folklore: It is discussed in Varga, for example. Since the power method has been discussed, it may be useful to discuss the convergence of CesÃ ro averages for irreducible nonnegative matrices; this may be due toKolmogorov.  Kiefer.Wolfowitz (talk) 12:41, 11 April 2010 (UTC)

text: ECONOMIC APPLICATIONS: I would like to add that Perron-Frobenius is very important in the macroeconomic analysis of the input-output model by Wassily Leontief. Specifically, it shows that a sector that takes inputs from other sectors but does not provide inputs to other sectors (e.g., jewelry making) causes the matrix to be reducible because it becomes a block to itself. In Leontief's model, each sector has a row and a column in the input-output matrix, so a(i,j) is the stuff made by sector i that is used by sector j; loadings (i.e., matrix values) are non-negative as if a sector does not trade with another sector, there will be a zero in the matrix. This may cause the matrix to become reducible. In addition to Leontief, a lot of work on this topic was made by Luigi Pasinetti. 161.239.232.131 (talk) 22:08, 15 March 2022 (UTC)
thread_title:  Irreducibility is required .... 

message: First let me thank contributors for their work (especially MRFS) at least one person (me) got a little bit happier from gaining knowledge from You. However there several things which seems to be not so clear at least for me. I will try to clarify by myself, but  Yours comments will be highly appreciated.
Alexander Chervov (talk) 19:08, 4 July 2010 (UTC)
replies: text: * Citation required: 
"A common thread in many proofs is the Brouwer fixed point theorem"
Alexander Chervov (talk) 19:08, 4 July 2010 (UTC)

text: * Is there definition of "spectral projection" on Wiki ?
"...spectral projection associated with the Perron root..."
Also material is a bit misleading because in theorem for positive matrices
by "Perron projection" it was called lim A^n/|r^n|. However such limit exists only for positive and primitive, not for generic irreducible - hence we have some implicit overloading the term Perron projection - first define by limit then redefine more generally as "spectral projection". 
I would suggest to avoid such overloadings or make it explicit.
Alexander Chervov (talk) 19:08, 4 July 2010 (UTC)

text: * Citation  and comment  required: 
"The Perron projection of an irreducible non-negative square matrix is a positive matrix".
Please provide citation, please comment on the proof, is it difficult or not ? Can one provide main idea ?Alexander Chervov (talk) 19:08, 4 July 2010 (UTC)

text: * Citation  and comment  required: 
" The key point is that a positive projection always has rank one. "
Please provide citation, please comment on the proof? Can one provide main idea ?Alexander Chervov (talk) 19:08, 4 July 2010 (UTC)

text: * Citation  and comment  required: 
"The power method is a convenient way to compute the Perron projection of a primitive matrix. If v and w are the positive row and column vectors that it generates then the Perron projection is just wv/vw."
Please provide citation, please comment on the proof? Can one provide main idea ?Alexander Chervov (talk) 19:08, 4 July 2010 (UTC)

text: * I just do not understand:
It should be noted that the spectral projections aren't neatly blocked as in the Jordan form. Here they are overlaid on top of one another and each generally has complex entries extending to all four corners of the square matrix.Alexander Chervov (talk) 19:08, 4 July 2010 (UTC)

text: * Citation  and comment  required: 
"So enter the peripheral projection which is the spectral projection of A corresponding to all the eigenvalues that have modulus Ï(A) ...."
Please provide citation, please comment on the definition of the "per. projection"? 
"The peripheral projection of an irreducible non-negative square matrix is a non-negative matrix with a positive diagonal."
Please provide citation, please comment on the proof? Can one provide main idea ?Alexander Chervov (talk) 19:08, 4 July 2010 (UTC)

text: * "Cyclicity" section - the same story
Alexander Chervov (talk) 19:08, 4 July 2010 (UTC)
thread_title:  Clarifications required 

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message: The section proving the uniqueness of an eigenvector with non-negative components relies on the claim that the eigenvectors corresponding to different eigenvalues are "orthogonal in some sense". It is not clear what is meant by this, meaning in what sense? Also, this claim is not justified or motivated at all in the article. How do we know that this is true? More clarification is needed. Thanks. Srfahmy (talk) 18:19, 31 January 2017 (UTC)
thread_title:  "Orthogonal in some sense" 

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message: The notions of left and right eigenvectors are switched throughout most of this page. This should be fixed.
128.250.0.120 (talk) 06:33, 22 September 2017 (UTC)
thread_title:  Left vs right eigenvector 

message: Conventional notation for the (i,j) element of matrix A is a_{ij}, not A_{ij}.  I am changing it in the article.  Zaslav (talk) 01:19, 15 August 2021 (UTC)
thread_title:  Matrix entry notation 

message: Does Perron–Frobenius hold for infinite square matrices? Thatsme314 (talk) 08:40, 2 October 2023 (UTC)
thread_title:  Infinite matrices