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| In [[linear algebra]], the '''Perron–Frobenius theorem''', proved by {{harvs|txt|authorlink=Oskar Perron|first=Oskar|last= Perron|year=1907}} and {{harvs|txt|authorlink=Georg Frobenius|first=Georg |last=Frobenius|year=1912}}, asserts that a real [[square matrix]] with positive entries has a unique largest real [[eigenvalue]] and that the corresponding [[eigenvector]] has strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory ([[ergodicity]] of [[Markov chain]]s); to the theory of [[dynamical systems]] ([[subshifts of finite type]]); to economics ([[Okishio's theorem]], [[Leontief]]'s [[input-output model]]);<ref name="Meyer681">{{harvnb|Meyer|2000|p=[http://www.matrixanalysis.com/Chapter8.pdf 8.3.6 p. 681]}}</ref>
| | == は 'まあ、最終的には神聖な木の無名の作品は少し奇妙ものの == |
| to demography ([[Leslie matrix|Leslie population age distribution model]]),<ref name="Meyer683">{{harvnb|Meyer|2000|p=[http://www.matrixanalysis.com/Chapter8.pdf 8.3.7 p. 683]}}</ref> to [[PageRank|Internet search engines]]<ref name="LangvilleMeyer167">{{harvnb|Langville|Meyer|2006|p=[http://books.google.com/books?id=hxvB14-I0twC&lpg=PP1&dq=isbn%3A0691122024&pg=PA167#v=onepage&q&f=false 15.2 p. 167]}}{{dead link|date=November 2012}}</ref> and even ranking of football
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| teams<ref name="Keener80">{{harvnb|Keener|1993|p=[http://links.jstor.org/sici?sici=0036-1445%28199303%2935%3A1%3C80%3ATPTATR%3E2.0.CO%3B2-O p. 80]}}</ref>
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| == Statement of the Perron–Frobenius theorem ==
| | 「これlinglongtaが大きい産業における私たちの絶妙な恵まれた神秘的な帝国である、私はあなたが大きな仲間の謎の帝国がそのような産業を置くかどうかを知らないのか？ [http://www.aseanacity.com/webalizer/prada-bags-24.html prada ベルト] '赤いマスクされた女性Piaoyunがパーティーを合図、座っていた漢も座る。<br>側は [http://www.aseanacity.com/webalizer/prada-bags-22.html prada トートバッグ] '何それ打开天窗说亮话物事、私は遠回しのが好きではありません。」、冷たい見えた<br>は [http://www.aseanacity.com/webalizer/prada-bags-22.html prada トートバッグ] 'まあ、最終的には神聖な木の無名の作品は少し奇妙ものの？ものですが、それは購入することを万能薬と中毒を購入するために多額の資金を持つ300万中毒価値はありません、仲間を教えてくださいこれはまた、別の学校の中で！元英ダンが多数に助成されているものが、最高の不死院英ダン、仲間すぎない弟子であり、彼らは不死のような多数どこから来た？ [http://www.aseanacity.com/webalizer/prada-bags-30.html プラダ ハンドバッグ] '赤女Piaoyun口をマスクした衣服は、一連の質問を吐き出す。<br>「これは、あなたが見つけるので、それは少し生意気なようで、私の**です [http://www.aseanacity.com/webalizer/prada-bags-30.html プラダ 財布]。「牙冷たい中空笑い大声で、「名前のない神のためとして、 |
| A [[matrix (mathematics)|matrix]] in which all entries are [[positive number|positive]] [[real number]]s is here called '''positive''' and a matrix whose entries are non-negative real numbers is here called '''non-negative'''. The [[eigenvalue]]s of a real [[square matrix]] ''A'' are [[complex numbers]] and collectively they make up the [[spectrum of a matrix|spectrum]] of the matrix. The [[exponential growth|exponential growth rate]] of the matrix powers ''A''<sup>''k''</sup> as ''k'' → ∞ is controlled by the eigenvalue of ''A'' with the largest [[absolute value]]. The Perron–Frobenius theorem describes the properties of the leading eigenvalue and of the corresponding eigenvectors when ''A'' is a non-negative real square matrix. Early results were due to {{harvs|txt|authorlink=Oskar Perron|first=Oskar|last= Perron|year=1907}} and concerned positive matrices. Later, {{harvs|txt|authorlink=Georg Frobenius|first=Georg |last=Frobenius|year=1912}} found their extension to certain classes of non-negative matrices.
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| === Positive matrices === | | == 精錬王丁へ == |
| Let <math>A = (a_{ij}) </math> be an <math> n \times n </math> positive matrix: <math> a_{ij} > 0 </math> for <math> 1 \le i,j \le n </math>. Then the following statements hold.
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| # There is a positive real number ''r'', called the '''Perron root''' or the '''Perron–Frobenius eigenvalue''', such that ''r'' is an eigenvalue of ''A'' and any other eigenvalue ''λ'' (possibly, [[complex number|complex]]) is strictly smaller than ''r'' in [[absolute value]], |''λ''| < ''r''. Thus, the [[spectral radius]] <math>\rho(A) </math> is equal to ''r''. If the matrix coefficients are algebraic, this implies that the eigenvalue is a [[Perron number]].
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| # The Perron–Frobenius eigenvalue is simple: ''r'' is a simple root of the [[characteristic polynomial]] of ''A''. Consequently, the [[eigenspace]] associated to ''r'' is one-dimensional. (The same is true for the left eigenspace, i.e., the eigenspace for ''A<sup>T</sup>''.)
| | 相关的主题文章： |
| # There exists an eigenvector ''v'' = (''v''<sub>1</sub>,…,''v''<sub>''n''</sub>) of ''A'' with eigenvalue ''r'' such that all components of ''v'' are positive: ''A v'' = ''r v'', ''v''<sub>''i''</sub> > 0 for 1 ≤ ''i'' ≤ ''n''. (Respectively, there exists a positive left eigenvector ''w'' : ''w<sup>T</sup> A'' = ''r w<sup>T</sup>'', ''w''<sub>''i''</sub> > 0.)
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| # There are no other positive (moreover non-negative) eigenvectors except positive multiples of ''v'' (respectively, left eigenvectors except ''w''), i.e., all other eigenvectors must have at least one negative or non-real component.
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| # <math> \lim_{k \rightarrow \infty} A^k/r^k = v w^T</math>, where the left and right eigenvectors for ''A'' are normalized so that ''w<sup>T</sup>v'' = 1. Moreover, the matrix ''v w<sup>T</sup>'' is the [[Jordan_canonical_form#Invariant_subspace_decompositions|projection onto the eigenspace]] corresponding to ''r''. This projection is called the '''Perron projection'''.
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| # '''[[Lothar Collatz|Collatz]]–Wielandt formula''': for all non-negative non-zero vectors ''x'', let ''f''(''x'') be the minimum value of [''Ax'']<sub>''i''</sub> / ''x''<sub>''i''</sub> taken over all those ''i'' such that ''x<sub>i</sub>'' ≠ 0. Then ''f'' is a real valued function whose [[maximum]] is the Perron–Frobenius eigenvalue.
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| # A "Min-max" Collatz–Wielandt formula takes a form similar to the one above: for all strictly positive vectors ''x'', let ''g''(''x'') be the maximum value of [''Ax'']<sub>''i''</sub> / ''x''<sub>''i''</sub> taken over ''i''. Then ''g'' is a real valued function whose [[minimum]] is the Perron–Frobenius eigenvalue.
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| # The Perron–Frobenius eigenvalue satisfies the inequalities
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| ::<math>\min_i \sum_{j} a_{ij} \le r \le \max_i \sum_{j} a_{ij}.</math>
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| These claims can be found in Meyer<ref name="Meyer"/> [http://www.matrixanalysis.com/Chapter8.pdf chapter 8] claims 8.2.11-15 page 667 and exercises 8.2.5,7,9 pages 668-669.
| | == negative side cold hands standing == |
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| The left and right eigenvectors ''v'' and ''w'' are usually normalized so that the sum of their components is equal to 1; in this case, they are sometimes called '''stochastic eigenvectors'''.
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| === Non-negative matrices === | | == air tumbling everywhere == |
| An extension of the theorem to matrices with non-negative entries is also available. In order to highlight the similarities and differences between the two cases the following points are to be noted: every non-negative matrix can be obviously obtained as a limit of positive matrices, thus one obtains the existence of an eigenvector with non-negative components; obviously the corresponding eigenvalue will be non-negative and greater '''or equal''' in absolute value than all other eigenvalues.<ref>{{harvnb|Meyer|2000|p=[http://www.matrixanalysis.com/Chapter8.pdf chapter 8.3 page 670]}}</ref><ref>{{harvnb|Gantmacher|2000|p=[http://books.google.com/books?id=cyX32q8ZP5cC&lpg=PA178&vq=preceding%20section&pg=PA66#v=onepage&q&f=false chapter XIII.3 theorem 3 page 66]}}</ref> However, the simple examples
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| : <math>
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| \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},
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| \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}</math>
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| show that for non-negative matrices there may exist eigenvalues of the same absolute value as the maximal one ((1) and (−1) – eigenvalues of the first matrix); moreover the maximal eigenvalue may not be a simple root of the characteristic polynomial, can be zero and the corresponding eigenvector (1,0) is not strictly positive (second example). So it may seem that most properties are broken for non-negative matrices, however Frobenius found the right way.
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| The key feature of theory in the non-negative case is to find some special subclass of non-negative matrices— '''irreducible''' matrices— for which a non-trivial generalization is possible. Namely, although eigenvalues attaining maximal absolute value may not be unique, the structure of maximal eigenvalues is under control: they have the form ''e''<sup>''i2πl/h''</sup>''r'', where ''h'' is some integer number—[[iterated function|period]] of matrix, ''r'' is a real strictly positive eigenvalue, ''l'' = 0, 1, ..., ''h'' − 1.
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| The eigenvector corresponding to ''r'' has strictly positive components (in contrast with the general case of non-negative matrices, where components are only non-negative). Also all such eigenvalues are simple roots of the characteristic polynomial. Further properties are described below.
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| ==== Classification of matrices ==== | | == more intense == |
| Let ''A'' be a square matrix (not necessarily positive or even real).
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| The matrix ''A'' is '''irreducible''' if any of the following equivalent properties
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| Definition 1 : ''A'' does not have non-trivial invariant '''coordinate''' subspaces.
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| Here a non-trivial coordinate subspace means a [[linear subspace]] spanned by any [[proper subset]] of '''basis''' vectors. More explicitly, for any linear subspace spanned by basis vectors ''e''<sub>''i''<sub>1</sub> </sub>, ...,
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| ''e''<sub>''i''<sub>k</sub></sub>, ''n'' > ''k'' > 0 its image under the action of ''A'' is not contained in the same subspace.
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| Definition 2: ''A'' cannot be conjugated into block upper triangular form by a [[permutation matrix]] ''P'':
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| : <math>PAP^{-1} \ne | | 相关的主题文章： |
| \begin{pmatrix} E & F \\ 0 & G \end{pmatrix},</math>
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| where ''E'' and ''G'' are non-trivial (i.e. of size greater than zero) square matrices.
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| If ''A'' is non-negative other definitions exist:
| | == side cold saw this woman == |
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| Definition 3: For every pair of indices ''i'' and ''j'', there exists a natural number ''m'' such that (''A''<sup>''m''</sup>)<sub>''ij''</sub> is not equal to 0.
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| Definition 4: One can associate with a matrix ''A'' a certain [[directed graph]] ''G''<sub>''A''</sub>. It has exactly ''n'' vertices, where ''n'' is size of ''A'', and there is an edge from vertex ''i'' to vertex ''j'' precisely when ''A''<sub>''ij''</sub> > 0. Then the matrix ''A'' is irreducible if and only if its associated graph ''G''<sub>''A''</sub> is [[strongly connected component|strongly connected]].
| | == I'm afraid will be some magic main plot == |
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| A matrix is '''reducible''' if it is not irreducible.
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| Let ''A'' be non-negative. Fix an index ''i'' and define the '''period of index ''i'' ''' to be the [[greatest common divisor]] of all natural numbers ''m'' such that (''A''<sup>''m''</sup>)<sub>''ii''</sub> > 0. When ''A'' is irreducible, the period of every index is the same and is called the '''period of ''A''. ''' In fact, when ''A'' is irreducible, the period can be defined as the greatest common divisor of the lengths of the closed directed paths in ''G''<sub>''A''</sub> (see Kitchens<ref name="Kitchens"/> page 16). The period is also called the index of imprimitivity
| | == Ying Han God monument == |
| (Meyer<ref name="Meyer"/> page 674) or the order of cyclicity.
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|
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| If the period is 1, ''A'' is '''aperiodic'''.
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| | | 相关的主题文章： |
| A matrix ''A'' is '''primitive''' if it is non-negative and its ''m''th power is positive for some natural number ''m'' (i.e. the same ''m'' works for all pairs of indices). It can be proved that primitive matrices are the same as irreducible aperiodic non-negative matrices.
| | <ul> |
| | | |
| A positive square matrix is primitive and a primitive matrix is irreducible. All statements of the Perron–Frobenius theorem for positive matrices remain true for primitive matrices. However, a general non-negative irreducible matrix ''A'' may possess several eigenvalues whose absolute value is equal to the spectral radius of ''A'', so the statements need to be correspondingly modified. Actually the number of such eigenvalues is exactly equal to the period. Results for non-negative matrices were first obtained by Frobenius in 1912.
| | <li>[http://www.carmtcpj.com/sns/space.php?uid=1838&do=blog&id=9734 http://www.carmtcpj.com/sns/space.php?uid=1838&do=blog&id=9734]</li> |
| | | |
| ==== Perron–Frobenius theorem for irreducible matrices ====
| | <li>[http://hao.liaohe.net.cn/plus/feedback.php?aid=3446 http://hao.liaohe.net.cn/plus/feedback.php?aid=3446]</li> |
| | | |
| Let ''A'' be an irreducible non-negative ''n'' × ''n'' matrix with period ''h'' and [[spectral radius]] ''ρ''(''A'') = ''r''. Then the following statements hold.
| | <li>[http://tjjames.com/cgi-bin/gb/guestbook.cgi http://tjjames.com/cgi-bin/gb/guestbook.cgi]</li> |
| | | |
| # The number ''r'' is a positive real number and it is an eigenvalue of the matrix ''A'', called the '''Perron–Frobenius eigenvalue'''.
| | </ul> |
| # The Perron–Frobenius eigenvalue ''r'' is simple. Both right and left eigenspaces associated with ''r'' are one-dimensional.
| |
| # ''A'' has a left eigenvector ''v'' with eigenvalue ''r'' whose components are all positive.
| |
| # Likewise, ''A'' has a right eigenvector ''w'' with eigenvalue ''r'' whose components are all positive.
| |
| # The only eigenvectors whose components are all positive are those associated with the eigenvalue ''r''.
| |
| # Matrix ''A'' has exactly ''h'' (where ''h'' is the '''period''') complex eigenvalues with absolute value ''r''. Each of them is a simple root of the characteristic polynomial and is the product of ''r'' with an ''h''th [[root of unity]].
| |
| # Let ''ω'' = 2π/''h''. Then the matrix ''A'' is [[similar matrix|similar]] to ''e''<sup>''iω''</sup>''A'', consequently the spectrum of ''A'' is invariant under multiplication by ''e''<sup>''iω''</sup> (corresponding to the rotation of the complex plane by the angle ''ω'').
| |
| # If ''h'' > 1 then there exists a permutation matrix ''P'' such that
| |
| ::<math>PAP^{-1}=
| |
| \begin{pmatrix}
| |
| 0 & A_1 & 0 & 0 & \ldots & 0 \\
| |
| 0 & 0 & A_2 & 0 & \ldots & 0 \\
| |
| \vdots & \vdots &\vdots & \vdots & & \vdots \\
| |
| 0 & 0 & 0 & 0 & \ldots & A_{h-1} \\
| |
| A_h & 0 & 0 & 0 & \ldots & 0
| |
| \end{pmatrix},
| |
| </math>
| |
| ::where the blocks along the main diagonal are zero square matrices.
| |
| | |
| :9. '''[[Lothar Collatz|Collatz]]–Wielandt formula''': for all non-negative non-zero vectors ''x'' let ''f''(''x'') be the minimum value of [''Ax'']<sub>''i''</sub> / ''x''<sub>''i''</sub> taken over all those ''i'' such that ''x<sub>i</sub>'' ≠ 0. Then ''f'' is a real valued function whose [[maximum]] is the Perron–Frobenius eigenvalue.
| |
| | |
| :10. The Perron–Frobenius eigenvalue satisfies the inequalities
| |
| ::<math>\min_i \sum_{j} a_{ij} \le r \le \max_i \sum_{j} a_{ij}.</math>
| |
| | |
| The matrix <math>\begin{pmatrix}0 & 0 & 1 \\0 & 0 & 1 \\1 & 1 & 0 \\\end{pmatrix}</math> shows that the blocks on the diagonal may be of different sizes, the matrices ''A''<sub>''j''</sub> need not be square, and ''h'' need not divide ''n''.
| |
| | |
| === Further properties ===
| |
| Let ''A'' be an irreducible non-negative matrix, then:
| |
| # (I+''A'')<sup>''n''−1</sup> is a positive matrix. (Meyer<ref name="Meyer"/> [http://www.matrixanalysis.com/Chapter8.pdf claim 8.3.5 p. 672]).
| |
| # Wielandt's theorem. If |''B''|<''A'', then ''ρ''(''B'')≤''ρ''(''A''). If equality holds (i.e. if ''μ=ρ(A)e<sup>iφ</sup>'' is eigenvalue for ''B''), then ''B'' = ''e''<sup>''iφ''</sup> ''D AD''<sup>−1</sup> for some diagonal unitary matrix ''D'' (i.e. diagonal elements of ''D'' equals to ''e''<sup>''iΘ''<sub>''l''</sub></sup>, non-diagonal are zero).<ref name="Meyer675">{{harvnb|Meyer|2000|p=[http://www.matrixanalysis.com/Chapter8.pdf claim 8.3.11 p. 675]}}</ref>
| |
| # If some power ''A<sup>q</sup>'' is reducible, then it is completely reducible, i.e. for some permutation matrix ''P'', it is true that: <math>
| |
| P A P^{-1}= \begin{pmatrix}
| |
| A_1 & 0 & 0 & \dots & 0 \\
| |
| 0 & A_2 & 0 & \dots & 0 \\
| |
| \vdots & \vdots & \vdots & & \vdots \\
| |
| 0 & 0 & 0 & \dots & A_d \\
| |
| \end{pmatrix}
| |
| </math>, where ''A<sub>i</sub>'' are irreducible matrices having the same maximal eigenvalue. The number of these matrices ''d'' is the greatest common divisor of ''q'' and ''h'', where ''h'' is period of ''A''.<ref>{{harvnb|Gantmacher|2000|p=section XIII.5 theorem 9}}</ref>
| |
| # If ''c(x)=x<sup>n</sup>+c<sub>k<sub>1</sub></sub> x<sup>n-k<sub>1</sub></sup> +c<sub>k<sub>2</sub></sub> x<sup>n-k<sub>2</sub></sup> + ... + c<sub>k<sub>s</sub></sub> x<sup>n-k<sub>s</sub></sup>'' is the characteristic polynomial of ''A'' in which the only nonzero coefficients are listed, then the period of ''A'' equals to the greatest common divisor for ''k<sub>1</sub>, k<sub>2</sub>, ... , k<sub>s</sub>''.<ref>{{harvnb|Meyer|2000|p=[http://www.matrixanalysis.com/Chapter8.pdf page 679]}}</ref>
| |
| # [[Cesàro summation|Cesàro]] [[summability theory|averages]]: <math> \lim_{k \rightarrow \infty} 1/k\sum_{i=0,...,k} A^i/r^i = ( v w^t),</math> where the left and right eigenvectors for ''A'' are normalized so that ''w''<sup>''t''</sup>''v'' = 1. Moreover the matrix ''v w<sup>t</sup>'' is the [[spectral projection]] corresponding to ''r'' - Perron projection.<ref>{{harvnb|Meyer|2000|p=[http://www.matrixanalysis.com/Chapter8.pdf example 8.3.2 p. 677]}}</ref>
| |
| # Let ''r'' be the Perron-Frobenius eigenvalue, then the adjoint matrix for (r-''A'') is positive.<ref>{{harvnb|Gantmacher|2000|p=[http://books.google.com/books?id=cyX32q8ZP5cC&lpg=PA178&vq=preceding%20section&pg=PA62#v=onepage&q&f=true section XIII.2.2 page 62]}}</ref>
| |
| # If ''A'' has at least one non-zero diagonal element, then ''A'' is primitive.<ref>{{harvnb|Meyer|2000|p= [http://www.matrixanalysis.com/Chapter8.pdf example 8.3.3 p. 678]}}</ref>
| |
| | |
| Also:
| |
| * If 0 ≤ ''A'' < ''B'', then ''r''<sub>''A''</sub> ≤ ''r''<sub>''B.''</sub> Moreover, if ''A'' is irreducible, then the inequality is strict: ''r<sub>A</sub> < r<sub>B</sub>''.
| |
| | |
| One of the definitions of primitive matrix requires ''A'' to be non-negative and there exists ''m'', such that ''A<sup>m</sup>'' is positive. One may one wonder how big ''m'' can be, depending on the size of ''A''. The following answers this question.
| |
| * Assume ''A'' is non-negative primitive matrix of size ''n'', then ''A<sup>n<sup>2</sup>-2n+2</sup>'' is positive. Moreover, if ''n'' > 1, there exists a matrix ''M'' given below, such that ''M<sup>k</sup>'' is not positive (but of course still non-negative) for all ''k < n<sup>2</sup>-2n+2'', in particular (''M''<sup>''n''<sup>2</sup>-2''n''+1</sup>)<sub>11</sub>=0.
| |
| :<math>M=
| |
| \begin{pmatrix}
| |
| 0 & 1 & 0 & 0 & ... & 0 \\
| |
| 0 & 0 & 1 & 0 & ... & 0 \\
| |
| 0 & 0 & 0 & 1 & ... & 0 \\
| |
| \vdots & \vdots & \vdots & \vdots & & \vdots \\
| |
| 0 & 0 & 0 & 0 & ... & 1 \\
| |
| 1 & 1 & 0 & 0 & ... & 0 \\
| |
| \end{pmatrix}
| |
| </math><ref>{{harvnb|Meyer|2000|p=[http://www.matrixanalysis.com/Chapter8.pdf chapter 8 example 8.3.4 page 679 and exercise 8.3.9 p. 685]}}</ref>
| |
| | |
| ==Applications==
| |
| Numerous books have been written on the subject of non-negative matrices, and Perron–Frobenius theory is invariably a central feature. The following examples given below only scratch the surface of its vast application domain.
| |
| | |
| ===Non-negative matrices===
| |
| The Perron–Frobenius theorem does not apply directly to non-negative matrices. Nevertheless any reducible square matrix ''A'' may be written in upper-triangular block form (known as the '''normal form of a reducible matrix''')<ref>{{harvnb|Varga|2002|p=2.43 (page 51)}}</ref>
| |
| ::::''PAP''<sup>−1</sup> = <math> \left( \begin{smallmatrix}
| |
| B_1 & * & * & \cdots & * \\
| |
| 0 & B_2 & * & \cdots & * \\
| |
| \vdots & \vdots & \vdots & & \vdots \\
| |
| 0 & 0 & 0 & \cdots & * \\
| |
| 0 & 0 & 0 & \cdots & B_h
| |
| \end{smallmatrix}
| |
| \right)</math>
| |
| | |
| where ''P'' is a permutation matrix and each ''B<sub>i</sub>'' is a square matrix that is either irreducible or zero. Now if ''A'' is non-negative then so are all the ''B<sub>i</sub>'' and the spectrum of ''A'' is just the union of their spectra. Therefore many of the spectral properties of ''A'' may be deduced by applying the theorem to the irreducible ''B<sub>i</sub>''.
| |
| | |
| For example the Perron root is the maximum of the ρ(''B<sub>i</sub>''). While there will still be eigenvectors with non-negative components it is quite possible that none of these will be positive.
| |
| | |
| ===Stochastic matrices===
| |
| A row (column) [[stochastic matrix]] is a square matrix each of whose rows (columns) consists of non-negative real numbers whose sum is unity. The theorem cannot be applied directly to such matrices because they need not be irreducible.
| |
| | |
| If ''A'' is row-stochastic then the column vector with each entry 1 is an eigenvector corresponding to the eigenvalue 1, which is also ρ(''A'') by the remark above. It might not be the only eigenvalue on the unit circle: and the associated eigenspace can be multi-dimensional. If ''A'' is row-stochastic and irreducible then the Perron projection is also row-stochastic and all its rows are equal.
| |
| | |
| ===Algebraic graph theory===
| |
| The theorem has particular use in [[algebraic graph theory]]. The "underlying graph" of a nonnegative ''n''-square matrix is the graph with vertices numbered 1, ..., ''n'' and arc ''ij'' if and only if ''A<sub>ij</sub>'' ≠ 0. If the underlying graph of such a matrix is strongly connected, then the matrix is irreducible, and thus the theorem applies. In particular, the [[adjacency matrix]] of a [[strongly connected component|strongly connected graph]] is irreducible.<ref>{{cite book |authorlink=Richard A. Brualdi |first=Richard A. |last=Brualdi |authorlink2=Herbert John Ryser |first2=Herbert John |last2=Ryser |title=Combinatorial Matrix Theory |location=Cambridge |publisher=Cambridge UP |year=1992 |isbn=0-521-32265-0 }}</ref><ref>{{cite book |authorlink=Richard A. Brualdi |first=Richard A. |last=Brualdi |first2=Dragos |last2=Cvetkovic |title=A Combinatorial Approach to Matrix Theory and Its Applications |publisher=CRC Press |location=Boca Raton, FL |year=2009 |isbn=978-1-4200-8223-4 }}</ref>
| |
| | |
| ===Finite Markov chains===
| |
| The theorem has a natural interpretation in the theory of finite [[Markov chain]]s (where it is the matrix-theoretic equivalent of the convergence of an irreducible finite Markov chain to its stationary distribution, formulated in terms of the transition matrix of the chain; see, for example, the article on the [[subshift of finite type]]).<!--which article?-->
| |
| | |
| ===Compact operators===
| |
| {{main|Krein–Rutman theorem}}
| |
| More generally, it can be extended to the case of non-negative [[compact operator]]s, which, in many ways, resemble finite-dimensional matrices. These are commonly studied in physics, under the name of [[transfer operator]]s, or sometimes '''Ruelle–Perron–Frobenius operators''' (after [[David Ruelle]]). In this case, the leading eigenvalue corresponds to the [[thermodynamic equilibrium]] of a [[dynamical system]], and the lesser eigenvalues to the decay modes of a system that is not in equilibrium. Thus, the theory offers a way of discovering the [[arrow of time]] in what would otherwise appear to be reversible, deterministic dynamical processes, when examined from the point of view of [[point-set topology]].<ref>{{cite book |first=Michael C. |last=Mackey |title=Time's Arrow: The origins of thermodynamic behaviour |location=New York |publisher=Springer-Verlag |year=1992 |isbn=0-387-97702-3 }}</ref>
| |
| | |
| ==Proof methods==
| |
| A common thread in many proofs is the [[Brouwer fixed point theorem]]. Another popular method is that of Wielandt (1950). He used the [[Lothar Collatz|Collatz]]–Wielandt formula described above to extend and clarify Frobenius's work.<ref>{{harvnb|Gantmacher|2000|p=[http://books.google.ru/books?id=cyX32q8ZP5cC&lpg=PR5&dq=Applications%20of%20the%20theory%20of%20matrices&pg=PA54#v=onepage&q&f=false section XIII.2.2 page 54]}}</ref> Another proof is based on the [[spectral theory]]<ref name="Smith"/> from which part of the arguments are borrowed.
| |
| | |
| ===Perron root is strictly maximal eigenvalue for positive (and primitive) matrices===
| |
| If ''A'' is a positive (or more generally primitive) matrix, then there exists a real positive eigenvalue ''r'' (Perron-Frobenius eigenvalue or Perron root), which is strictly greater in absolute value than all other eigenvalues, hence ''r'' is the [[spectral radius]] of ''A''.
| |
| | |
| This statement does not hold for general non-negative irreducible matrices, which have ''h'' eigenvalues with the same absolute eigenvalue as ''r'', where ''h'' is the period of ''A''.
| |
| | |
| ====Proof for positive matrices====
| |
| Let ''A'' be a positive matrix, assume that its spectral radius ρ(''A'') = 1 (otherwise consider ''A/ρ(A)''). Hence, there exists an eigenvalue λ on the unit circle, and all the other eigenvalues are less or equal 1 in absolute value. Assume that λ ≠ 1. Then there exists a positive integer ''m'' such that ''A<sup>m</sup>'' is a positive matrix and the real part of λ''<sup>m</sup>'' is negative. Let ε be half the smallest diagonal entry of ''A<sup>m</sup>'' and set ''T'' = ''A<sup>m</sup>'' − ''ε''1 which is yet another positive matrix. Moreover if ''Ax'' = λ''x'' then ''A<sup>m</sup>x'' = ''λ<sup>m</sup>x'' thus λ''<sup>m</sup>'' − ''ε'' is an eigenvalue of ''T''. Because of the choice of ''m'' this point lies outside the unit disk consequently ρ(''T'') > 1. On the other hand all the entries in ''T'' are positive and less than or equal to those in ''A<sup>m</sup>'' so by [[spectral radius|Gelfand's formula]] ρ(''T'') ≤ ρ(''A<sup>m</sup>'') ≤ ρ(''A)<sup>m</sup>'' = 1. This contradiction means that λ=1 and there can be no other eigenvalues on the unit circle.
| |
| | |
| Absolutely the same arguments can be applied to the case of primitive matrices, after one just need to mention the following simple lemma, which clarifies the properties of primitive matrices
| |
| | |
| ====Lemma====
| |
| Given a non-negative ''A'', assume there exists ''m'', such that ''A<sup>m</sup>'' is positive, then ''A<sup>m+1</sup>, A<sup>m+2</sup>, A<sup>m+3</sup>,...'' are all positive.
| |
| | |
| ''A<sup>m+1</sup>= A A<sup>m</sup>'', so it can have zero element only if some row of ''A'' is entirely zero, but in this case the same row of ''A<sup>m</sup>'' will be zero.
| |
| | |
| Applying the same arguments as above for primitive matrices, prove the main claim.
| |
| | |
| ===Power method and the positive eigenpair===
| |
| For a positive (or more generally irreducible non-negative) matrix ''A'' the dominant [[eigenvector]] is real and strictly positive (for non-negative ''A'' respectively non-negative.)
| |
| | |
| This can be established using the [[power method]], which states that for a sufficiently generic (in the sense below) matrix ''A'' the sequence of vectors ''b''<sub>''k''+1</sub>=''Ab''<sub>''k''</sub> / | ''Ab''<sub>''k''</sub> | converges to the [[eigenvector]] with the maximum [[eigenvalue]]. (The initial vector ''b<sub>0</sub>'' can be chosen arbitrarily except for some measure zero set). Starting with a non-negative vector ''b<sub>0</sub>'' produces the sequence of non-negative vectors ''b<sub>k</sub>''. Hence the limiting vector is also non-negative. By the power method this limiting vector is the dominant eigenvector for ''A'', proving the assertion. The corresponding eigenvalue is non-negative.
| |
| | |
| The proof requires two additional arguments. First, the power method converges for matrices which do not have several eigenvalues of the same absolute value as the maximal one. The previous section's argument guarantees this.
| |
| | |
| Second, to ensure strict positivity of all of the components of the eigenvector for the case of irreducible matrices.
| |
| This follows from the following fact, which is of independent interest:
| |
| | |
| :Lemma: given a positive (or more generally irreducible non-negative) matrix ''A'' and ''v'' as any non-negative eigenvector for ''A'', then it<!--the eigenvector?--> is necessarily strictly positive and the corresponding eigenvalue is also strictly positive.
| |
|
| |
| Proof. One of the definitions of irreducibility for non-negative matrices is that for all indexes ''i,j'' there exists ''m'', such that (''A''<sup>''m''</sup>)<sub>''ij''</sub> is strictly positive. Given a non-negative eigenvector ''v'', and that at least one of its components say ''j''-th is strictly positive, the corresponding eigenvalue is strictly positive, indeed, given ''n'' such that (''A''<sup>''n''</sup>)<sub>''ii''</sub> >0, hence: ''r''<sup>''n''</sup>''v''<sub>''i''</sub> =
| |
| ''A''<sup>''n''</sup>''v''<sub>''i''</sub> >=
| |
| (''A''<sup>''n''</sup>)<sub>''ii''</sub>''v''<sub>''i''</sub>
| |
| >0. Hence ''r'' is strictly positive. The eigenvector is strict positivity. Then given ''m'', such that (''A''<sup>''m''</sup>)<sub>''ij''</sub> >0, hence: ''r''<sup>''m''</sup>''v''<sub>''j''</sub> =
| |
| (''A''<sup>''m''</sup>''v'')<sub>''j''</sub> >=
| |
| (''A''<sup>''m''</sup>)<sub>''ij''</sub>''v''<sub>''i''</sub> >0, hence
| |
| ''v''<sub>''j''</sub> is strictly positive, i.e., the eigenvector is strictly positive.
| |
| | |
| === Multiplicity one ===
| |
| The proof that the Perron-Frobenius eigenvalue is a simple root of the characteristic polynomial is also elementary. The arguments here are close to those in Meyer.<ref name="Meyer">{{harvnb|Meyer|2000|p=[http://www.matrixanalysis.com/Chapter8.pdf chapter 8 page 665]}}</ref> Hence the eigenspace associated to Perron-Frobenius eigenvalue ''r'' is one-dimensional.
| |
| | |
| Given a strictly positive eigenvector ''v'' corresponding to ''r'' and another eigenvector ''w'' with the same eigenvalue. (Vector ''w'' can be chosen to be real, because ''A'' and ''r'' are both real, so the null space of ''A-r'' has a basis consisting of real vectors). Assuming at least one of the components of ''w'' is positive (otherwise multiply ''w'' by -1). Given maximal possible ''α'' such that ''u=v- α w'' is non-negative, then one of the components of ''u'' is zero, otherwise ''α'' is not maximum. Vector ''u'' is an eigenvector. It is non-negative, hence by the lemma described in the [[Perron–Frobenius_theorem#Power_method_and_the_positive_eigenpair|previous section]] non-negativity implies strict positivity for any eigenvector. On the other hand as above at least one component of ''u'' is zero. The contradiction implies that ''w'' does not exist.
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| | |
| Case: There are no Jordan cells corresponding to the Perron-Frobenius eigenvalue ''r'' and all other eigenvalues which have the same absolute value.
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| If there is a Jordan cell, then the [[Matrix_norm#Induced_norm|infinity norm]]
| |
| (A/r)<sup>k</sup><sub>∞</sub> tends to infinity for ''k → ∞ '',
| |
| but that contradicts the existence of the positive eigenvector.
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| | |
| Given ''r=1'', or ''A/r''. Letting ''v'' be a Perron-Frobenius strictly positive eigenvector, so ''Av=v'', then:
| |
| | |
| <math> \|v\|_{\infty}= \|A^k v\|_{\infty} \ge \|A^k\|_{\infty} \min_i (v_i), ~~\Rightarrow~~ \|A^k\|_{\infty} \le \|v\|/\min_i (v_i) </math>
| |
| So ''A<sup>k</sup>''<sub>∞</sub> is bounded for all ''k''. This gives another proof that there are no eigenvalues which have greater absolute value than Perron-Frobenius one. It also contradicts the existence of the Jordan cell for any eigenvalue which has absolute value equal to 1 (in particular for the Perron-Frobenius one), because existence of the Jordan cell implies that ''A<sup>k</sup>''<sub>∞</sub> is unbounded. For a two by two matrix:
| |
| : <math>
| |
| J^k= \begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix} ^k
| |
| =
| |
| \begin{pmatrix} \lambda^k & k\lambda^{k-1} \\ 0 & \lambda^k \end{pmatrix},
| |
| </math>
| |
| hence ''J<sup>k</sup>''<sub>∞</sub> = ''|k+λ|'' (for ''|λ|=1''), so it tends to infinity when ''k'' does so. Since ''J<sup>k</sup>'' = ''C<sup>-1</sup> A<sup>k</sup>C'' , then '' A<sup>k</sup>'' >= ''J<sup>k</sup>''/ ( ''C''<sup>−1</sup> ''C'' ), so it also tends to infinity. The resulting contradiction implies that there are no Jordan cells for the corresponding eigenvalues.
| |
| | |
| Combining the two claims above reveals that the Perron-Frobenius eigenvalue ''r'' is simple root of the characteristic polynomial. In the case of nonprimitive matrices, there exist other eigenvalues which have the same absolute value as ''r''. The same claim is true for them, but requires more work.
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| | |
| === No other non-negative eigenvectors ===
| |
| Given positive (or more generally irreducible non-negative matrix) ''A'', the Perron-Frobenius eigenvector is the only (up to multiplication by constant) non-negative eigenvector for ''A''.
| |
| | |
| Other eigenvectors should contain negative, or complex components. Since eigenvectors for different eigenvalues are orthogonal in some sense, but two positive eigenvectors cannot be orthogonal, so they must correspond to the same eigenvalue, but the eigenspace for the Perron-Frobenius is one dimensional.
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| | |
| Assuming there exists an eigenpair ''(λ, y)'' for ''A'', such that vector ''y'' is positive, and given ''(r, x)'', where ''x'' - is the right Perron-Frobenius eigenvector for ''A'' (i.e. eigenvector for ''A<sup>t</sup>''), then
| |
| ''r x<sup>t</sup> y = (x<sup>t</sup> A) y= x<sup>t</sup> (A y)=λ x<sup>t</sup> y'', also ''x<sup>t</sup> y >0'', so one has: '' r=λ''. Since the eigenspace for the Perron-Frobenius eigenvalue ''r'' is one dimensional, non-negative eigenvector ''y'' is a multiple of the Perron-Frobenius one.<ref>{{harvnb|Meyer|2000|p=[http://www.matrixanalysis.com/Chapter8.pdf chapter 8 claim 8.2.10 page 666]}}</ref>
| |
| | |
| === Collatz–Wielandt formula ===
| |
| Given a positive (or more generally irreducible non-negative matrix) ''A'', for all non-negative non-zero vectors ''x'' and ''f''(''x'') as the minimum value of [''Ax'']<sub>''i''</sub> / ''x''<sub>''i''</sub> taken over all those ''i'' such that ''x<sub>i</sub>'' ≠ 0, then ''f'' is a real valued function whose [[maximum]] is the Perron–Frobenius eigenvalue ''r''.
| |
| | |
| Here, ''r'' is attained for ''x'' taken to be the Perron-Frobenius eigenvector ''v''. The proof requires that values ''f'' on the other vectors are less or equal. Given a vector ''x''. Let ''ξ=f(x)'', so ''0<= ξx <=Ax'' and ''w'' to be the right eigenvector for ''A'', then ''w<sup>t</sup> ξx <= w<sup>t</sup> (Ax) = (w<sup>t</sup> A)x = r w<sup>t</sup> x ''. Hence ξ<=r.<ref>{{harvnb|Meyer|2000|p=[http://www.matrixanalysis.com/Chapter8.pdf chapter 8 page 666]}}</ref>
| |
| | |
| ===Perron projection as a limit: ''A<sup>k</sup>/r<sup>k</sup>''===
| |
| Let ''A'' be a positive (or more generally, primitive) matrix, and let ''r be'' its Perron-Frobenius eigenvalue.
| |
| # There exists a limit ''A<sup>k</sup>/r<sup>k</sup>'' for ''k → ∞'', denote it by ''P''.
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| # ''P'' is a [[Projection (linear algebra)|projection operator]]: ''P<sup>2</sup>=P'', which commutes with ''A'': ''AP=PA''.
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| # The image of ''P'' is one-dimensional and spanned by the Perron-Frobenius eigenvector ''v'' (respectively for ''P<sup>t</sup>''—by the Perron-Frobenius eigenvector ''w'' for ''A<sup>t</sup>'').
| |
| # ''P= v w<sup>t</sup>'', where ''v,w'' are normalized such that ''w<sup>t</sup> v'' = 1.
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| # Hence ''P'' is a positive operator.
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|
| |
| Hence ''P'' is a [[spectral projection]] for the Perron-Frobenius eigenvalue ''r'', and is called the Perron projection.
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| The above assertion is not true for general non-negative irreducible matrices.
| |
| | |
| Actually the claims above (except claim 5) are valid for any matrix ''M'' such that there exists an eigenvalue ''r'' which is strictly greater than the other eigenvalues in absolute value and is the simple root of the characteristic [[polynomial]]. (These requirements hold for primitive matrices as above).
| |
| | |
| Given that ''M'' is diagonalizable, M is conjugate to a diagonal matrix with eigenvalues ''r<sub>1</sub>, ... , r<sub>n</sub>'' on the diagonal (denote ''r<sub>1</sub>=r''). The matrix ''M<sup>k</sup>/r<sup>k</sup>'' will be conjugate ''(1, (r<sub>2</sub>/r)<sup>k</sup>, ... , (r<sub>n</sub>/r)<sup>k</sup>)'', which tends to (1,0,0,...,0), for ''k → ∞'', so the limit exists. The same method works for general ''M'' (without assuming that ''M'' is diagonalizable).
| |
| | |
| The projection and commutativity properties are elementary corollaries of the definition: ''M M<sup>k</sup>/r<sup>k</sup>= M<sup>k</sup>/r<sup>k</sup> M'' ; ''P<sup>2</sup> = lim M<sup>2k</sup>/r<sup>2k</sup>=P''. The third fact is also elementary: '' M (P u)= M lim M<sup>k</sup>/r<sup>k</sup> u = lim r M<sup>k+1</sup>/r<sup>k+1</sup> u'', so taking the limit yields ''M (P u)=r (P u)'', so image of ''P'' lies in the ''r''-eigenspace for ''M'', which is one-dimensional by the assumptions.
| |
| | |
| Denoting by ''v'', ''r''-eigenvector for ''M'' (by ''w'' for ''M<sup>t</sup>''). Columns of ''P'' are multiples of ''v'', because the image of ''P'' is spanned by it. Respectively, rows of ''w''. So ''P'' takes a form ''(a v w<sup>t</sup>)'', for some ''a''. Hence its trace equals to ''(a w<sup>t</sup> v)''. Trace of projector equals the dimension of its image. It was proved before that it is not more than one-dimensional. From the definition one sees that ''P'' acts identically on the ''r''-eigenvector for ''M''. So it is one-dimensional. So choosing ''(w<sup>t</sup> v)=1'', implies ''P=v w<sup>t</sup>''.
| |
| | |
| ===Inequalities for Perron–Frobenius eigenvalue===
| |
| For any non-nonegative matrix ''A'' its Perron–Frobenius eigenvalue ''r'' satisfies the inequality:
| |
| | |
| :<math> r \; \le \; \max_i \sum_j A_{ij}.</math>
| |
| | |
| This is not specific to non-negative matrices: for any matrix ''A'' with eigenvalue ''λ'' it is true
| |
| that <math>\scriptstyle |\lambda| \; \le \; \max_i \sum_j |A_{ij}|.</math>. This is an immediate corollary of the
| |
| [[Gershgorin circle theorem]]. However another proof is more direct:
| |
| | |
| Any [[Matrix_norm#Induced_norm|matrix induced norm]] satisfies the inequality ''A''λ| for any eigenvalue ''λ'', because ''A'' ≥ ''Ax''/x = λ''x''/x = |λ|. The [[Matrix_norm#Induced_norm|infinity norm]] of a matrix is the maximum of row sums: <math>\scriptstyle \left \| A \right \| _\infty = \max \limits _{1 \leq i \leq m} \sum _{j=1} ^n | a_{ij} |. </math> Hence the desired inequality is exactly ''A''λ| applied to non-negative matrix ''A''.
| |
| | |
| Another inequality is:
| |
| | |
| :<math>\min_i \sum_j A_{ij} \; \le \; r .</math>
| |
| | |
| This fact is specific to non-negative matrices; for general matrices there is nothing similar. Given that ''A'' is positive (not just non-negative), then there exists a positive eigenvector ''w'' such that ''Aw'' = ''rw'' and the smallest component of ''w'' (say ''w<sub>i</sub>'') is 1. Then ''r'' = (''Aw'')<sub>''i''</sub> ≥ the sum of the numbers in row ''i'' of ''A''. Thus the minimum row sum gives a lower bound for ''r'' and this observation can be extended to all non-negative matrices by continuity.
| |
| | |
| Another way to argue it is via the [[Lothar Collatz|Collatz]]-Wielandt formula. One takes the vector ''x'' = (1, 1, ..., 1) and immediately obtains the inequality.
| |
| | |
| ===Further proofs===
| |
| | |
| ====Perron projection====
| |
| The proof now proceeds using [[spectral decomposition]]. The trick here is to split the Perron root from the other eigenvalues. The spectral projection associated with the Perron root is called the Perron projection and it enjoys the following property:
| |
| | |
| The Perron projection of an irreducible non-negative square matrix is a positive matrix.
| |
| | |
| Perron's findings and also (1)–(5) of the theorem are corollaries of this result. The key point is that a positive projection always has rank one. This means that if ''A'' is an irreducible non-negative square matrix then the algebraic and geometric multiplicities of its Perron root are both one. Also if ''P'' is its Perron projection then ''AP'' = ''PA'' = ρ(''A'')''P'' so every column of ''P'' is a positive right eigenvector of ''A'' and every row is a positive left eigenvector. Moreover if ''Ax'' = λ''x'' then ''PAx'' = λ''Px'' = ρ(''A'')''Px'' which means ''Px'' = 0 if λ ≠ ρ(''A''). Thus the only positive eigenvectors are those associated with ρ(''A''). If ''A'' is a primitive matrix with ρ(''A'') = 1 then it can be decomposed as ''P'' ⊕ (1 − ''P'')''A'' so that ''A<sup>n</sup>'' = ''P'' + (1 − ''P'')''A''<sup>''n''</sup>. As ''n'' increases the second of these terms decays to zero leaving ''P'' as the limit of ''A<sup>n</sup>'' as ''n'' → ∞.
| |
| | |
| 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''. It should be noted that the spectral projections aren't neatly blocked as in the Jordan form. Here they are overlaid and each generally has complex entries extending to all four corners of the square matrix. Nevertheless they retain their mutual orthogonality which is what facilitates the decomposition.
| |
| | |
| ====Peripheral projection====
| |
| The analysis when ''A'' is irreducible and non-negative is broadly similar. The Perron projection is still positive but there may now be other eigenvalues of modulus ρ(''A'') that negate use of the power method and prevent the powers of (1 − ''P'')''A'' decaying as in the primitive case whenever ρ(''A'') = 1. So we consider the '''peripheral projection''', which is the spectral projection of ''A'' corresponding to all the eigenvalues that have modulus ''ρ''(''A''). It may then be shown that the peripheral projection of an irreducible non-negative square matrix is a non-negative matrix with a positive diagonal.
| |
| | |
| ====Cyclicity====
| |
| Suppose in addition that ρ(''A'') = 1 and ''A'' has ''h'' eigenvalues on the unit circle. If ''P'' is the peripheral projection then the matrix ''R'' = ''AP'' = ''PA'' is non-negative and irreducible, ''R<sup>h</sup>'' = ''P'', and the cyclic group ''P'', ''R'', ''R''<sup>2</sup>, ...., ''R''<sup>''h''−1</sup> represents the harmonics of ''A''. The spectral projection of ''A'' at the eigenvalue λ on the unit circle is given by the formula <math>\scriptstyle h^{-1}\sum^h_1\lambda^{-k}R^k</math>. All of these projections (including the Perron projection) have the same positive diagonal, moreover choosing any one of them and then taking the modulus of every entry invariably yields the Perron projection. Some donkey work is still needed in order to establish the cyclic properties (6)–(8) but it's essentially just a matter of turning the handle. The spectral decomposition of ''A'' is given by ''A'' = ''R'' ⊕ (1 − ''P'')''A'' so the difference between ''A<sup>n</sup>'' and ''R<sup>n</sup>'' is ''A<sup>n</sup>'' − ''R<sup>n</sup>'' = (1 − ''P'')''A''<sup>''n''</sup> representing the transients of ''A<sup>n</sup>'' which eventually decay to zero. ''P'' may be computed as the limit of ''A<sup>nh</sup>'' as ''n'' → ∞.
| |
| | |
| ==Caveats==
| |
| The matrices ''L'' = <math>\left(
| |
| \begin{smallmatrix}
| |
| 1 & 0 & 0 \\
| |
| 1 & 0 & 0 \\
| |
| 1 & 1 & 1
| |
| \end{smallmatrix}
| |
| \right)</math>, ''P'' = <math>\left(
| |
| \begin{smallmatrix}
| |
| \;\;\;1 & 0 & 0 \\
| |
| \;\;\;1 & 0 & 0 \\
| |
| -1 & 1 & 1
| |
| \end{smallmatrix}
| |
| \right)</math>, ''T'' = <math>\left(
| |
| \begin{smallmatrix}
| |
| 0 & 1 & 1 \\
| |
| 1 & 0 & 1 \\
| |
| 1 & 1 & 0
| |
| \end{smallmatrix}
| |
| \right)</math>, ''M'' = <math>\left(
| |
| \begin{smallmatrix}
| |
| 0 & 1 & 0 & 0 & 0 \\
| |
| 1 & 0 & 0 & 0 & 0 \\
| |
| 0 & 0 & 0 & 1 & 0 \\
| |
| 0 & 0 & 0 & 0 & 1 \\
| |
| 0 & 0 & 1 & 0 & 0
| |
| \end{smallmatrix}
| |
| \right)</math> provide simple examples of what can go wrong if the necessary conditions are not met. It is easily seen that the Perron and peripheral projections of ''L'' are both equal to ''P'', thus when the original matrix is reducible the projections may lose non-negativity and there is no chance of expressing them as limits of its powers. The matrix ''T'' is an example of a primitive matrix with zero diagonal. If the diagonal of an irreducible non-negative square matrix is non-zero then the matrix must be primitive but this example demonstrates that the converse is false. ''M'' is an example of a matrix with several missing spectral teeth. If ω = e<sup>iπ/3</sup> then ω<sup>6</sup> = 1 and the eigenvalues of ''M'' are {1,ω<sup>2</sup>,ω<sup>3</sup>,ω<sup>4</sup>} so ω and ω<sup>5</sup> are both absent.{{Citation needed|date=January 2012}}
| |
| | |
| ==Terminology==
| |
| A problem that causes confusion is a lack of standardisation in the definitions. For example, some authors use the terms ''strictly positive'' and ''positive'' to mean > 0 and ≥ 0 respectively. In this article ''positive'' means > 0 and ''non-negative'' means ≥ 0. Another vexed area concerns ''decomposability'' and ''reducibility'': ''irreducible'' is an overloaded term. For avoidance of doubt a non-zero non-negative square matrix ''A'' such that 1 + ''A'' is primitive is sometimes said to be ''connected''. Then irreducible non-negative square matrices and connected matrices are synonymous.<ref>For surveys of results on irreducibility, see [[Olga Taussky-Todd]] and [[Richard A. Brualdi]].</ref>
| |
| | |
| The nonnegative eigenvector is often normalized so that the sum of its components is equal to unity; in this case, the eigenvector is the vector of a [[probability distribution]] and is sometimes called a ''stochastic eigenvector''.
| |
| | |
| ''Perron–Frobenius eigenvalue'' and ''dominant eigenvalue'' are alternative names for the Perron root. Spectral projections are also known as ''spectral projectors'' and ''spectral idempotents''. The period is sometimes referred to as the ''index of imprimitivity'' or the ''order of cyclicity''.
| |
| | |
| == See also ==
| |
| | |
| * [[Z-matrix (mathematics)]]
| |
| * [[M-matrix]]
| |
| * [[P-matrix]]
| |
| * [[Hurwitz matrix]]
| |
| * [[Metzler matrix]] ([[Quasipositive matrix]])
| |
| * [[Positive operator]]
| |
| | |
| == Notes ==
| |
| {{reflist|3|refs=
| |
| | |
| <ref name="Kitchens">{{citation | title=Symbolic dynamics: one-sided, two-sided and countable state markov shifts. | year=1998 | first1=Bruce | last1=Kitchens | url = http://books.google.ru/books?id=mCcdC_5crpoC&lpg=PA195&ots=RbFr1TkSiY&dq=kitchens%20perron%20frobenius&pg=PA16#v=onepage&q&f=false|publisher = Springer
| |
| }}</ref>
| |
| <ref name="Smith">{{citation | title = A Spectral Theoretic Proof of Perron–Frobenius| pages = 29–35| volume = 102 | number = 1 | journal =Mathematical Proceedings of the Royal Irish Academy | year=2006 | first1=Roger | last1=Smith | url = ftp://emis.maths.adelaide.edu.au/pub/EMIS/journals/MPRIA/2002/pa102i1/pdf/102a102.pdf |publisher = The Royal Irish Academy| doi=10.3318/PRIA.2002.102.1.29
| |
| }}</ref>
| |
| }}
| |
| | |
| == References ==
| |
| | |
| === Original papers ===
| |
| *{{Citation | last1=Perron | first1=Oskar | title=Zur Theorie der Matrices | doi=
| |
| 10.1007/BF01449896 | year=1907 | journal= Mathematische Annalen | volume=64
| |
| | issue=2 | pages=248–263 | author-link = Oskar Perron}}
| |
| | |
| *{{citation|first=Georg|last= Frobenius|title=Ueber Matrizen aus nicht negativen Elementen|journal= Sitzungsber. Königl. Preuss. Akad. Wiss. |year=1912|pages=456–477 | author-link = Georg Frobenius }}
| |
| | |
| *{{citation|first=Georg|last= Frobenius|title=Über Matrizen aus positiven Elementen, 1 |journal= Sitzungsber. Königl. Preuss. Akad. Wiss. |year=1908|pages=471–476 | author-link = Georg Frobenius}}
| |
| | |
| *{{citation|first=Georg|last= Frobenius|title=Über Matrizen aus positiven Elementen, 2 |journal= Sitzungsber. Königl. Preuss. Akad. Wiss. |year=1909|pages=514–518 | author-link = Georg Frobenius }}
| |
| | |
| * {{citation | title= The Theory of Matrices, Volume 2 | origyear = 1959| year= 2000 | first1=Felix | last1=Gantmacher | publisher = AMS Chelsea Publishing| author1-link=Felix Gantmacher| isbn= 0-8218-2664-6|
| |
| url = http://books.google.com/books?id=cyX32q8ZP5cC&lpg=PA178&vq=preceding%20section&pg=PA53#v=onepage&q&f=true}} (1959 edition had different title: "Applications of the theory of matrices". Also the numeration of chapters is different in the two editions.)
| |
| | |
| *{{citation | title=Google page rank and beyond | year=2006 | first1=Amy | last1=Langville | first2=Carl | last2=Meyer | url = http://pagerankandbeyond.com |publisher = Princeton University Press | ISBN = 0-691-12202-4
| |
| }}{{dead link|date=November 2012}}
| |
| | |
| *{{citation | title=The Perron–Frobenius theorem and the ranking of football teams | year=1993 | first1=James | last1=Keener | url = http://links.jstor.org/sici?sici=0036-1445%28199303%2935%3A1%3C80%3ATPTATR%3E2.0.CO%3B2-O|publisher = SIAM | journal = SIAM Review| volume = 35| number =1| pages = 80–93
| |
| }}
| |
| | |
| *{{citation | title=Matrix analysis and applied linear algebra | year=2000 | first1=Carl | last1=Meyer | url = http://www.matrixanalysis.com/Chapter8.pdf|publisher = SIAM | isbn = 0-89871-454-0
| |
| }}
| |
| *{{citation|first=V.|last= Romanovsky|title=Sur les zéros des matrices stocastiques
| |
| |journal= Bulletin de la Société Mathématique de France |year=1933|pages=213–219
| |
| | volume = 61 | url = http://www.numdam.org/item?id=BSMF_1933__61__213_0 }}
| |
| | |
| *{{citation|first=Lothar |last= Collatz| author-link = Lothar Collatz|title=Einschließungssatz für die charakteristischen Zahlen von Matrize
| |
| |journal= Mathematische Zeitschrift |year=1942|pages=221–226
| |
| |doi = 10.1007/BF01180013| volume = 48 | number =1 }}
| |
| | |
| *{{citation|first=Helmut|last= Wielandt|title=Unzerlegbare, nicht negative Matrizen
| |
| |journal= Mathematische Zeitschrift |year=1950|pages=642–648
| |
| |doi = 10.1007/BF02230720| volume = 52 | number =1 }}
| |
| | |
| === Further reading ===
| |
| | |
| * Abraham Berman, Robert J. Plemmons, ''Nonnegative Matrices in the Mathematical Sciences'', 1994, SIAM. ISBN 0-89871-321-8.
| |
| * [[Chris Godsil]] and [[Gordon Royle]], ''Algebraic Graph Theory'', Springer, 2001.
| |
| * A. Graham, ''Nonnegative Matrices and Applicable Topics in Linear Algebra'', John Wiley&Sons, New York, 1987.
| |
| * R. A. Horn and C.R. Johnson, ''Matrix Analysis'', Cambridge University Press, 1990
| |
| * S. P. Meyn and R. L. Tweedie, [https://netfiles.uiuc.edu/meyn/www/spm_files/book.html ''Markov Chains and Stochastic Stability''] London: Springer-Verlag, 1993. ISBN 0-387-19832-6 (2nd edition, Cambridge University Press, 2009)
| |
| *Henryk Minc, ''Nonnegative matrices'', John Wiley&Sons, New York, 1988, ISBN 0-471-83966-3
| |
| * Seneta, E. ''Non-negative matrices and Markov chains''. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Series in Statistics. (Originally published by Allen & Unwin Ltd., London, 1973) ISBN 978-0-387-29765-1
| |
| *{{eom|id=P/p072350|first=D.A. |last=Suprunenko}} (The claim that ''A''<sub>''j''</sub> has order ''n''/''h'' at the end of the statement of the theorem is incorrect.)
| |
| * [[Richard S. Varga]], ''Matrix Iterative Analysis'', 2nd ed., Springer-Verlag, 2002
| |
| | |
| {{DEFAULTSORT:Perron-Frobenius Theorem}}
| |
| [[Category:Matrix theory]]
| |
| [[Category:Theorems in linear algebra]]
| |
| [[Category:Mathematical and quantitative methods (economics)]]
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| [[Category:Markov processes]]
| |
は 'まあ、最終的には神聖な木の無名の作品は少し奇妙ものの
「これlinglongtaが大きい産業における私たちの絶妙な恵まれた神秘的な帝国である、私はあなたが大きな仲間の謎の帝国がそのような産業を置くかどうかを知らないのか？ prada ベルト '赤いマスクされた女性Piaoyunがパーティーを合図、座っていた漢も座る。
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は prada トートバッグ 'まあ、最終的には神聖な木の無名の作品は少し奇妙ものの？ものですが、それは購入することを万能薬と中毒を購入するために多額の資金を持つ300万中毒価値はありません、仲間を教えてくださいこれはまた、別の学校の中で！元英ダンが多数に助成されているものが、最高の不死院英ダン、仲間すぎない弟子であり、彼らは不死のような多数どこから来た？ プラダ ハンドバッグ '赤女Piaoyun口をマスクした衣服は、一連の質問を吐き出す。
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a 'semi elders' among speak, took out a bottle of immortality, 'This is five pieces of refraining Dan, can make you maintain good physical strength in battle demons you want to take or not to take the good.'
'Nature does not quit.' immortality took a cold side, the pace of this ancient Shimen entered into.
this pedal out into the Shimen of the square prada トートバッグ seems to feel cold on into the dark abyss, dizzy, almost all turn his head dizzy, then
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more intense
Tattoos on his body, but prada ピンク 財布 also live up to God diagram evolved a picture of St. maps, road maps, rotate around the whole body.
palm strike out, even in the body of the prada ピンク 財布 force ròu, Ying Han Fang good fortune Boxer cold thirty-three days, thirty-three times the combat power.
Bang! Two shares prada ベルト of god in プラダ 長財布 the central core of the barbarian army to fight, a プラダ 財布 迷彩 huge storm that swept out of the formation of hundreds of millions of dragon storm, destroyed the barbarians and outsiders do not know how many soldiers.
systemic side cold cracking, who appeared dense porcelain general rift, shocking. Wu holy ghost figure are about playing volley, from his side.
and that the barbarian saint main military penetration of the blood, bloody tattoos on the body, but his face sè heavy, but the enormous body of breath, more intense, seems to be injured but makes him excited, this really is not a saint statue so easy to beat, cast out in the cold side fighting force thirty times
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side cold saw this woman
These Taikoo Shing pools, all of them are fetish combined together prada ベルト to form a country, almost all of the country among all the glory, order, disaster, chaos, eternal, tragedy, people robbed ......... 財布 ブランド プラダ even further have good fortune, prada 財布 リボン the origin, the truth ...... pass all sorts of myths which includes Vientiane, seems to have some cold era and square way similar.
ninety-nine eighty-one presented the king of the city, putting the repression, that master of yin and yang fall immediately hit the mundane, the body exploded, of yin and yang to the gas again.
'yin and yang, to close!'
in the city that Hong eighty-one king, presents a huge statue of a woman ghost that float along a white woman, prada トートバッグ sits プラダ 財布 定価 Zhonggu, forehead are all among a cold smile.
'Kiyoshi!'
side cold saw this woman, my heart could not help but burst of excitement, because this woman is the mother of Heaven's Soldiers electricity
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I'm afraid will be some magic main plot
He did not have any of the move, sweating. Just a hint for their ideas feel very stupid.
'You are the master Yuanshi magic of the magic number of people, a very high position, so I'll give you a task, capitulate main spring and autumn, came to my door era, I can give him a law enforcement position, if not, I have overcome the whole Yuanshi magic 財布 プラダ レディース number. primary Blood River, the main tomb of the Lord against the demons, Hadean Lord that I have been beheaded, the magic number is left now Yuanshi how many demons Lord, give me amnesty , which prada 財布 2014 is a great credit, you know that? '
party commanded a プラダ レディース 財布 hint of cold.
'yes! door Lord, I will settle the matter.' Chi Wang again lower his noble head, but he then had some hesitation, 'the main door, I just promoted to reach Heaven's Soldiers realm of magic still shallow go Yuanshi magic number amnesty, I'm afraid will be some magic main plot, 財布 ブランド プラダ unable town Da pressed them now Yuanshi magic number prada メンズ 財布 in addition to the spring
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Ying Han God monument
Shrouded magic hands, suddenly appeared in the six objectives monument Gods above, Ying Han God monument, on the potential of a shot!
Bang! Six God
monument, click on the photograph in the battle to be running slow up! Wang took the opportunity to blood pradaの財布 the night in which destroy the moving force, carnage soaring! Piercing out: 'Blood Rain 長財布 プラダ sky!' Bang bang bang bang bang bang
!
six mesh Gods monument was Zhenfei!
shrouded magic in hand, under the blood of the king of the night together, six head Gods monument Pizhen open! prada ピンク 財布 King of プラダ ピンク 財布 blood the night, after all, Dao, and shrouded magic hand, it is magical ten heavy ghost Emperor lifelong trained in magic hand, especially when the magic hands which combines shrouded Mogong, and in side cold excel Five Emperors big devil promotion supernatural sixfold Normalized territory when cold blood and square integration, more powerful!
two baby force forceful. Six mesh Gods monument after all, not Dao, it is impossible to fully control the プラダ 財布 新作 2014 magic hands all night and Wang blood repression live!
'shrouded magic hands!
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