Talk:Perron–Frobenius theorem

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)

I thought positivity of a matrix was related to its eigenvalues, not to its matrix entries? â Preceding unsigned comment added by 145.30.124.13 (talk) 13:23, 5 February 2013 (UTC)

Definition
these are just some mistakes etc etc etc.... user: ÙØšØªØ¯ØŠ
 * 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)
 * Hello there are some littel mistakes in the article i think. I corriged them but another user eraesed my changes:
 * i just want to say that positive means >=0 and exsit at least one element not rqual to zero
 * strictly positiv >0 means there is no elemnt of the vector or Matrix equal to 0
 * non negative means >=0 that means for example the zero vektor too
 * furthermore only the sum of the columns in a stochastic matrix is one and not the rows


 * 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 "&ge;" or something like that, and I'm not sure right now which is right. Michael Hardy 17:55, 26 September 2006 (UTC)


 * 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 "&ge; 0". Michael Hardy 17:57, 26 September 2006 (UTC)

The matrix (0 1 // 1 0) seems to be a counterexample to the statement of the Perron-Frobenius theorem for non-negative matrices as stated in this article. The eigenvalues of this matrix are 1 and -1, and
 * -1| is not strictly less than 1. -- Dave Benson, Aberdeen, 13 Feb 2007.


 * 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)


 * This page would suggest that a non-strict inequality applies in the non-negative case. Rawling 4851 09:59, 15 February 2007 (UTC)

if one assumes that the entries of the matrix are only non-negative (which, for applications, is often the right hypothesis), then one needs extra assumptions for the theorem to be valid: irreducibility (to ensure that all entries of the eigenvector are (strictly) positive) and primitiveness (to avoid other eigenvalues with the same modulus).


 * Just want to add in the case of non-negative matrices the spectral radius may not necessarily be an eigenvalue - but if not, then its negative must be an eigenvalue. For example, think about -I where I is the identity matrix. I've amended the text. âPreceding unsigned comment added by 193.129.26.250 (talk) 14:54, 1 July 2008 (UTC)

94.254.42.40 (talk) 09:24, 26 March 2010 (UTC)
 * The argument for existence of a strictly positive eigenvector and eigenvalue seems flawed to me... Doesn't using the power method assume an eigenvalue of larger magnitude than all others? What happens when we don't have this?


 * Good point. There's a big gap in the argument. The power method can't be applied until it has been shown that the spectral radius has larger magnitude than the other eigenvalues. MRFS (talk) 09:09, 30 March 2010 (UTC)


 * One just need to "pass by continuity". Just one sentence. Arguments are restored. Any way, thank You guys ! Alexander Chervov (talk) 16:20, 4 July 2010 (UTC)

Irreducibility is required ....
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)


 * 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)


 * 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)


 * 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)

First of all thank you to KW and MH for taking the rewrite on board.

Naturally I realise that some advance warning would have been appropriate, but I suspect not many people read the TALK pages. Moreover it's clear from this particular one that quite a few people were having difficulty understanding the theorem. I was tempted to post, "Does anybody agree with me that it's time this article was rewritten?", but refrained from doing so because I felt it would have attracted any number of different answers.

The main sources are Dunford & Schwartz, Linear Operators I (for the spectral theory) and Marcus & Minc, A survey of matrix theory and matrix inequalities, which states the theorem without proof. My own aim was to find a proof that offered rather more insight than the traditional ones into why the theorem actually worked, and for that reason existing proofs were deliberately NOT used. The search proved fruitless until the properties of the various spectral projections began to emerge. The technical details may be found in "A spectral theoretic proof of Perron-Frobenius", Mathematical Proceedings of the Royal Irish Academy, 102A (1), 29-35 (2002).

I described the canonical form as "folklore" because although it's essential for the non-negative application I've never seen it properly attributed to anyone. I've seen it wrongly described as "Frobenius normal form", but as it's easy to prove I think it hardly needs any attribution.

The power method is incidental to the proof. I mentioned it only because of the anonymous post above a fortnight ago by 94.254.42.40 pointing out (quite rightly) that it is only applicable to primitive matrices. That was what prompted my unilateral action in the form of "imposing a total rewrite", for which I do apologise. MRFS (talk) 11:08, 12 April 2010 (UTC)


 * 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)

Clean-up
I have cleaned up the section on positive matrices, largely by restoring an earlier version that was mathematically correct and concise. I have also restored the lead that gives the actual statement of the theorem, as opposed to handwaving ("states results about <&hellip;>"). The section that is really bothering me now is "Towards proofs". I don't think that Wikipedia is a proper venue for giving insights into technical proofs, and especially, for piecing together various relevant but disparate facts that go into a proof. Some of the issues are WP:OR and WP:UNDUE , but there are also pragmatic considerations: maintenance of proofs in an open wiki project is difficult.


 * Wikipedia allows proofs under restriction of being not too technical. See WikiProject Mathematics/Proofs. There is even a category "Articles containing proofs" with about 380 pages therein.


 * The rule WP:OR  - original research - so does not prohibit to provide   standard proofs.  The rule   WP:UNDUE - undue weight - is not applicable here - main idea of "undue" is not to present views of minorities as standard viewpoint, so has nothing to do here.  Only one sentence in this rule  "depth of detail" can be applied here. So first it is itself "undue weight" to apply the rule which 99% it is not applicable here to this situation. Second ideas of proofs often weights much more than theorems themselves, big portion of mathematics contains  of proofs - and this is emphasized in  WikiProject Mathematics/Proofs says: "...Because much of the published professional literature of mathematics consists of the details of proofs, it would be very difficult to write in any depth about mathematics without including at least some proofs or proof sketches..."Alexander Chervov (talk) 14:11, 4 July 2010 (UTC)


 * User MRFS says below "I agree that Wikipedia is definitely not a proper venue for technical proofs, but there will certainly be readers who are interested in why the theorem works and it seems unfair to deny them insight into why non-negative matrices behave the way they do."


 * Great words. Cannot say better. Alexander Chervov (talk) 09:41, 4 July 2010 (UTC).

There is actually a very short conceptual proof of the positive case based on the power method and contraction mapping theorem, and, perhaps, it could be inserted in the article, but anything beyond that isn't likely to be helpful. Arcfrk (talk) 00:56, 26 April 2010 (UTC)


 * Would you be so kind to insert or provide a citation ? Alexander Chervov (talk) 18:51, 6 July 2010 (UTC)

Some clean-up! If this article is allowed to stand in it's present form it will be back to square one - confusing, inaccurate in places, and rarely concise. Let's get this straight. The Perron-Frobenius theorem refers specifically to irreducible non-negative square matrices. It says nothing (as is claimed in the intro) about positive matrices except as special cases. It says nothing (as is claimed in the first section) about non-negative matrices in general.


 * Yes, it does! I don't have a copy of Lind and Marcus close at hand, but that's at least one source which states the theorem separately for positive matrices and non-negative matrices. As far as I remember, that is fairly common.

Statement of the Perron-Frobenius theorem :  The statement is conspicuous in absentia! Nowhere is there anything to say, "Here is the PF theorem". The rate of growth of Ak is not controlled by the eigenvalue of A with the largest absolute value. Let A be the identity matrix (1 0 // 0 1) and B = (1 1 // 0 1). In both cases the dominant eigenvalue is 1 but the powers of A are bounded whereas those of B are unbounded.


 * You misunderstand conventions of Wikipedia. The two subsections, "Positive matrices" and "Non-negative matrices" give the statements of the theorem for the corresponding cases.

Positive matrices :  It is inappropriate to define the Perron root under the heading "Positive matrices" since it applies (more generally) to all non-negative matrices. Likewise statement (5) is unnecessarily restrictive.


 * Even so, it needs to be defined here in order to state the result.

Non-negative matrices :  Here we go again. There is no such thing as the "Perron-Frobenius theorem for positive matrices". The results referred to were due to Perron alone. The Perron-Frobenius eigenvalue is (confusingly) defined a second time. Under (7) it is a nonsense to claim that A is similar to eiÏA because the spectrum is invariant under rotation through Ï. This implication is simply the wrong way round.


 * Many mathematical results are not historically named. How about Pell's equation?

Discussion :  In its present form the article is neither mathematically correct nor concise. As for the section entitled "Towards proofs" it was introduced by Alexander Chervov on 8 August 2009 and it remained unchanged despite an obvious flaw that was eventually pointed out by 94.254.42.40 on 26 March (see above).


 * One just need to "pass by continuity". Just one sentence. Arguments are restored. Any way thank You guys. Alexander Chervov (talk) 16:20, 4 July 2010 (UTC)

I agree that Wikipedia is definitely not a proper venue for technical proofs, but there will certainly be readers who are interested in why the theorem works and it seems unfair to deny them insight into why non-negative matrices behave the way they do. The section contains no WP:OR and criticism on grounds of WP:UNDUE seems hard to sustain since matrices and linear operators go hand in hand and the latter are intimately tied in with spectral theory. Indeed the spectral theory of linear operators on general Banach spaces points the way to results on countably infinite matrices as mentioned in the Generalizations section. So the message to readers who may not like this section is that they don't have to read it. A proof (as suggested) of Perron's results alone would be quite inadequate since the main meat of the theorem lies in the work of Frobenius.


 * Most of these complaints equally apply to the previous version. I've fixed or removed many inaccurate, biased, and/or misleading statements, but I don't have an opportunity at the moment to check the present text against a good source, so I cannot certify, for example, that the statement of PF theorem for irreducible matrices is mathematically correct or standard. It's a fact of life that there are incorrect statements scattered over Wikipedia that stay around for a long time, even though people realize that they are wrong: it may not be worthwhile to fix one thing if the article is generally in poor shape.

Conclusion :  The Moderators should review the recent drafts of this article and decide which one to retain. MRFS (talk) 15:14, 26 April 2010 (UTC)


 * There are no moderators on Wikipedia, but I've left a note at the Math Project page, so, hopefully, some people will take a look. Arcfrk (talk) 20:31, 26 April 2010 (UTC)


 * Regarding "Think it gooder": Mathematical concepts and language keep evolving. It is true that Perron's original work considered positive matrices. However, a lot of reliable and authoritative mathematicians often refer to the "Perron-Frobenius Theorem" in the general sense of nonnegative matrices.
 * IMHO, Wikipedia is more of a newspaper account of the front-lines of language purification, namely Mathematical Reviews, Linear Algebra and its Application, SIAM-Matrix, and Seminaire Bourbaki! If USA's Supreme-Court Justices Scalia and Thomas join us and push their original intent doctrine on Wikipedia, so much the worse for WP!
 * Let's have one article on the positive dominant eigenvector of nonnegative matrices, rather than one for Perron, one for Frobenius, one for Kolmogorov, one for von Neumann, one for Stephen M. Robinson,  etc. Thanks! Kiefer.Wolfowitz (talk) 00:16, 27 April 2010 (UTC)


 * The question isn't whether the non-negative case is called PF theorem, it's the opposite: does the positive case deserve separate treatment? I've just consulted three standard recent books on dynamical systems, Clark Robinson, Bruce Kitchens, and KatokâHasselblatt, and all of them formulate the special case for primitive matrices (in the terminology of this article) separately and call it "PerronâFrobenius theorem" (with slight variations). As I had mentioned earlier, LindâMarcus does the same. Thus this use is supported by the literature. There is a definite pedagogical advantage in treating the more restrictive case (positive matrices) independently. Moreover, for some applications, the easier case is all one needs and the result is sharper. I agree that both cases should be discussed in a single article. On the other hand, I am not sure what Kolmogorov's and von Neumann's contributions to PF theory were, but if they go beyond what standard sources call "PerronâFrobenius theorem" then most certainly they need to be described in a different article or articles. Arcfrk (talk) 00:41, 27 April 2010 (UTC)
 * Von Neumann extended PF theory to rectangular matrix pencils $$p^T (\mathbf{A}-\lambda \mathbf{B}) q= 0$$ for nonnegative nonzero matrices $$\mathbf{A,B}$$ and for stochastic vectors $$p$$ and $$q$$, introducing the so-called Collatz-Wielandt function (OR,btw) and using a "whales and wranglers" requirement of (too strong) irreducibility. Robinson gave the appropriate definition of irreducibility (Econometrica c. 1974). See Gerald Thompson and Oskar Morgenstern's Mathematical Theory of Expanding and Contracting Economies, Schrijver's book on linear programming, etc., or the article by Thompson in SIAM Review in the early 1970s, symposia of the Polish Academy of Sciences (edited by the couple Los); extensions include Rockafellar's theory of monotone processes of convex and concave type and extensions to set valued processes by Rubinov and Makarov and J.P. Aubin, Robinson (again), Jon Borwein (see corrections by Urescu or Zalinescu), etc. Kiefer.Wolfowitz (talk) 09:10, 30 April 2010 (UTC)

I don't disagree with anything that is said here but the following points are relevant.

First of all I know it's common for "reliable and authoritative mathematicians" to refer to the P-F theorem in the sense of non-negative matrices but that's because they know what they're talking about! For the uninitiated (as is very evident from the posts above) the distinctions between positive, primitive, and irreducible matrices can be highly confusing. Therefore it is essential that this article is crystal clear on the context of each result.

Secondly is there is any advantage to be gained by treating the different types of matrix independently? In the interests of clarity the answer has to be an emphatic "yes". Moreover in most applications the matrices are primitive at worst, so the irreducible case is mainly of theoretical interest to pure mathematicians, many of whom regard it as the most beautiful part of the theorem.

Thirdly should all the different types of matrix be discussed in a single article? Once again there are compelling reasons why they should be. Some things like the Perron root / P-F eigenvalue and the bounds on it in terms of row and column sums apply to all non-negative matrices, so a single article is needed to put everything in the right context.

Fourthly as well as providing "a newspaper account of the front-lines of language purification" there is an opportunity for WP to "think it gooder" in several respects. Your view of the P-F theory is likely to be slightly different depending on whether your interest is in dynamical systems or Markov processes or information technology or spectral theory or whatever. Typically each discipline has its own definitions (which can be a massive source of confusion) so this article offers a chance to reconcile these different views.

Rightly or wrongly I felt that up to the end of March the early sections of the article dated back several years and were no longer fit for purpose. There wasn't even a full statement of the theorem! In addition someone had pointed out a glaring mistake in the section "Towards a proof" but seemingly nothing was being done about it. So I produced a rewrite on 11 April which (to my eye) meets all these criteria. For the current draft to largely reinstate those early sections seems to me a definite backward step. MRFS (talk) 11:27, 27 April 2010 (UTC)

The referenced article by Suprunenko in the Encyclopaedia of Mathematics gives a reasonable statement of the theorem right up to its final line where it goes off the rails in claiming that h is a factor of n. The matrix ( 0 1 1 0 // 0 0 0 1 // 0 0 0 1 // 4 0 0 0 ) has eigenvalues 0, 2Ï, 2Ï2, 2 where Ï is a complex cube root of unity so clearly h = 3 and n = 4. MRFS (talk) 12:34, 1 May 2010 (UTC)

Clean up (cont'd)
I have pared down the reference list to a subset consisting of books that could potentially be used in sourcing this article (it still appears to be too long) and replaced a rambling section on "Generalizations" with a short "See also" list of related topics. Arcfrk (talk) 08:21, 24 May 2010 (UTC)


 * 1) In its present position the statement on exponential growth rates is simply wrong (since it doesn't apply to non-negative matrices in general). I'm moving it to the irreducible section.
 * 2) I'm adding a minor clarification to the reducible/irreducible definition.
 * 3) I've written to the Editor of the Encyclopaedia of Mathematics requesting the statement of the theorem be fixed. Standard reference or not I would be strongly against using it in the meantime. MRFS (talk) 20:42, 27 May 2010 (UTC)


 * I don't understand your obsession with eliminating the uncontroversial but important interpretation of the leading eigenvalue of a matrix. The standard definition, which unfortunately doesn't seem to be clearly enunciated on wikipedia, is that a non-negative sequence an has exponential growth rate r if $$\lim_{n\to\infty}\sqrt[n]{a_n}=r.$$ If a k by k matrix A has eigenvalues &lambda;1,&hellip;,&lambda;k then the norm of its nth power is between C1|&lambda;|n and C2n|&lambda;|n, where C1 and C2 are constants independent of n and &lambda; is the eigenvalue with the largest absolute value (all norms are equivalent and you can conjugate A to the Jordan normal form and explicitly compute the power). What exactly are you objecting to? Arcfrk (talk) 00:26, 28 May 2010 (UTC)


 * We're slightly at cross purposes. I think if you look here Gelfand's formula you will find the well known formula that you say isn't clearly enunciated on Wikipedia. It applies to a far wider field than the non-negative matrices of interest to P-F. It hardly needs restated here though I certainly would never object to it as it is a central plank in operator theory, Banach algebras, etc.


 * What I thought you were driving at is that the P-F theorem provides us with a much stronger interpretation in the case of irreducible non-negative matrices. Basically if such a matrix T has P-F eigenvalue 1 then its powers are bounded and there will be h limit points. However this fails for reducible matrices. Take the very simple example T = (1 1 // 0 1) which has P-F eigenvalue 1 and put any matrix norm you like on it. Gelfand's formula tells us that the limit of ||Tk||1/k is 1 but it doesn't follow from this that the sequence Tk is bounded. In fact this sequence is unbounded which shows that the stronger interpretation only works in the case of irreducible matrices. So I was objecting to what I saw as the stronger interpretation being presented out of context and before irreducible matrices had even been defined!


 * I hope this clears the matter up. Sincere apologies for any confusion. MRFS (talk) 09:52, 28 May 2010 (UTC)


 * Well, I don't see exponential growth rate of a sequence defined there (which indicates that you once again misread what I'd written), but yes, that page explains why the statement is true. I think that for most people, "exponential growth rate" is easy to grasp and doesn't have the strong connotation of being asymptotic to the exponential function; in any event, for positive matrices the stronger statement is a fortiori true. I've restored the remark to its proper place in the beginning. Arcfrk (talk) 00:21, 29 May 2010 (UTC)


 * Yes, as soon as I had posted I realised you were alluding to the lack of a definition of "exponential growth rate" rather than Gelfand's formula. But once the "Save" button is pressed the post is gone and there's no way of recalling it. So I shall just retrieve the ball from the back of the net and play on.


 * The original statement claimed that "The rate of growth of the matrix powers Ak as k â â is controlled by the eigenvalue of A with the largest absolute value". There was no mention of "exponential", no indication of what was meant by "control", and the "eigenvalue of A with the largest absolute value" is just a complex number that may not even be unique. So this statement was vague, misleading, and inaccurate and its removal was justified since my simple example showed that "linear growth" is not necessarily controlled by this eigenvalue. Sadly the current version only addresses one of my three objections. I suggest it should say that "The exponential growth rate of the matrix powers Ak as k â â is equal to the spectral radius". Then its meaning would be unambiguous though some people might still wonder why it is deemed necessary to include a remark on general matrices in a topic specific to non-negative ones. We might as well throw in a description of the Jordan form ....


 * So I did not misread the original statement but after I had deleted it and something similar immediately appeared in its place I was, frankly, so exasperated that I didn't bother to read it properly and missed the word "exponential". That was misread #2. Whilst linear growth is easy to understand I suspect very few people think of growth in terms of norms and nth roots and (as you say) Wikipedia does not clearly enunciate the standard definition of exponential growth. Why don't you put this right?


 * Finally here are a few thoughts that I must stress are not directed at yourself or indeed any particular editor of Wiki. I believe we will all achieve better results if we work in a spirit of cooperation rather than confrontation. Petty criticism benefits nobody and is just a waste of time. Indulging in reversion wars is equally pointless. Personally I see no value in referencing external articles (eg Supruenko) that contain nothing beyond what is already in Wiki. Even standard texts are not always 100% reliable and it will be interesting to see how long the Encyclopedia of Mathematics takes to correct its own version of the PF theorem.


 * I cannot agree with you more that Wiki is a cooperative project and it works best in that mode. Your exasperation is understandable, but we'll all do better if we assume good faith. Several remarks: (1) You can always edit your comments (many people do!), cross out parts of them by using, or even revert yourself. (2) The word "exponential" had indeed been missing and I was lazy about not adding it (my rationale: anyone who can follow the text can either fill it in mentally or decide that it's significant and edit the text, it's a wiki!) (3) I completely disagree with the philosophy of making precise statements with all the caveats in the explanatory text: this is what the statements of the theorems are for. For someone without a firm grasp of the material, the details can really be overwhelming (this is largely true even in mathematics monographs, but absolutely essential in survey and encyclopaedia articles), and what you object to as "vague" is a very helpful plain language explanation to many others. If some technical detail only becomes important in Section 100, that's where it needs to be pointed out. (4) Adding growth rate material to WP seems like a good idea, since you believe it is so important, why don't you do it? (I have my share of projects where I think I can be more efficient and that I can enjoy more.)(5) EOM is a standard resource, and as such, it's absence is conspicuous. R.e.b's solution is very good in that it takes care of maintenance issues and accuracy issues at the same time. I am way over my time allowance, so I hope you'll excuse me for putting an end to this discussion. See you around! Arcfrk (talk) 23:06, 30 May 2010 (UTC)

Irreducible aperiodic versus Primitive
I've just spotted a recent addition under "classification of matrices" which suggests the irreducible aperiodic non-negative matrices are a larger class than the primitive ones. That's not true. If A belongs to the former and its Perron root is 1 with Perron projection P then since 1 is simple its spectral decomposition is A = P â R where Ï(R) < 1. Therefore Ak â P as k â â and as P is positive Ak is ultimately positive which implies A is primitive. Surely this characterisation of primitive matrices (ie those with h = 1) is well known? MRFS (talk) 10:17, 30 May 2010 (UTC)


 * First of all I would like to you and to the other contributors for the excellent work you have done with this article. Just a couple of minor points as I'm not an expert on this topic. In the article you referred to as "classification of matrices", it seems to me to be a too general title. Even thought that it is into the "Non-negative matrices section", I think it would be useful for non expert public to find a more specific title, otherwise it looks it is the only kind of classification scheme for these matrices.
 * The second thing is that some sections are not completely autonomous regarding notation. This is one of such a section. When you say for instance: "for any linear subspace spanned by basis vectors  ei 1, ... ,ei k , n > k > 0",  the meanings of n and k are not explained. Even if one can guess the meaning reading the whole article I think it should be autonomous in order to facilitate reading for those who access directly to this section. This happens also at the first point in the "Further properties" section, and maybe in others, but I didn't check the whole article looking for these points then I only realise when I didn't follow it quickly. I don't get the point neither for the subindex i in the definition of the basis vectors.
 * Thanks in advance. Best. Conjugado (talk) 17:42, 22 February 2011 (UTC)

Clarifications required
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)


 * Citation required:

"A common thread in many proofs is the Brouwer fixed point theorem" Alexander Chervov (talk) 19:08, 4 July 2010 (UTC)


 * 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)


 * 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)


 * 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)


 * 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)


 * 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)


 * 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)


 * "Cyclicity" section - the same story

Alexander Chervov (talk) 19:08, 4 July 2010 (UTC)

What is described under "Proof methods" is just a sketch of the proof which omits the technical details. My original post made this clear but various editors have hacked at it in the meantime with the result that much of the explanation has been removed.

Since Wikipedia is mainly for reference the inclusion of this section is open to question. The main motivation for writing it was because (in my view) the spectral theory approach is the best one for providing insight into why the theorem actually works, and that will certainly interest at least some of the readership. The full text of the underlying paper may be found here. ftp://emis.maths.adelaide.edu.au/pub/EMIS/journals/MPRIA/2002/pa102i1/pdf/102a102.pdf

Hopefully it will answer most of the above points. The rest can be picked up in due course. MRFS (talk) 12:11, 5 July 2010 (UTC)

Holomorphic functional calculus?
I changed spectral projection to projection onto the invariant subspace associated with the spectral radius, linking to its subsection on the invariant-subspace decomposition.

This article concerns matrix theory, not Banach algebras, so a link to holomorphic functional calculus was fancy-schmancy overkill, made worse by its failure to which could have been improved by linking to the spectral projection subsection Holomorphic_functional_calculus.

A previous version of this article had more references on Boas-Pringsheim's theorem (e.g., Karlin, Schaeffer) and on positive operators on Riesz spaces, but that material was removed. That subsection might have been an appropriate place for linking to functional calculi. Kiefer.Wolfowitz (talk) 14:03, 7 February 2011 (UTC)


 * Well, that's less concise as there are many projections onto the eigenspace associated with r. The object in question is the one onto the generalized eigenspace associated with r and along all other generalized eigenspaces, in other words the spectral projection at r.


 * The original link to holomorphic functional calculus was not "fancy-schmancy overkill". I didn't like it myself - the title is probably enough to put some people off - but it is the natural home on Wiki for spectral projections. In the introduction it makes clear that it is concerned with bounded linear operators on Banach spaces. It only mentions Banach algebras as an aside near the end so I fail to see why you deem it more relevant to them than to matrices. As for my not linking to the relevant subsection, I'm afraid you will just have to put up with casual readers like myself who are less familiar with Wiki than you are.


 * A very minor revision to Jordan canonical form would accommodate spectral projections so perhaps that's the best solution. MRFS 14:27, 8 February 2011 (UTC)
 * You are correct. I am sorry about my tone yesterday. Sincerely, Kiefer.Wolfowitz  (talk) 14:46, 8 February 2011 (UTC)
 * No problem. And thanks for the useful info on editing Wiki! MRFS (talk) 10:32, 9 February 2011 (UTC)

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"Orthogonal in some sense"
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)

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Left vs right eigenvector
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)

Matrix entry notation
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)

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