Math 19b: Linear Algebra with Probability Oliver Knill, Spring 2011Lecture 34: Perron Frobenius theoremThis is a second lecture on Markov processes. We want to see why the following result is true:If all entries of a Markov matrix A are p ositive then A has a unique equilibrium:there is only one eigenvalue 1. All other eigenvalues are smaller than 1.To illustrate the importance of the result, we look how it is used in chaos theory and how it canbe used fo r search engines to rank pages.1 The matrixA ="1/2 1/31/2 2/3#is a Markov matrix for which all entries are positive. The eigenvalue 1 is unique because thesum of the eigenvalues is 1/2 + 2/3 < 2.2 We have already proven Perron-Frobenius for 2 × 2 Markov matrices: such a matr ix is ofthe formA ="a b1 − a 1 − b#and has an eigenvalue 1 and a second eigenvalue smaller than 1 because tr(A) the sum ofthe eigenvalues is smaller than 2.3 Lets give a brute force proo f of the Perron-Frobenius theorem in the case of 3 × 3 matrices:such a matrix is of the formA =a b cd e f1 − a − d 1 − b − e 1 − c − f.and has an eigenvalue 1. The determinant is D = c(d − e) + a(e − f ) + b(−d + f ) and isthe product of the two remaining eigenvalues. The trace is 1 + (a − c) + (e − f ) so thatT = (a−c)+(e − f) is the sum of the two remaining eigenvalues. An ugly verification showsthat these eigenvalues are in absolute value smaller tha n 1.The Markov assumption is actually not needed. Here is a more general statement which is usefulin other parts mathematics. It is also one the theorems with the most applications like Leontief’smodels in economics, chaos theory in dynamical systems or page rank for search engines.Perron Frobenius theorem: If all entries of a n × n matrix A are positive, thenit has a unique maximal eigenvalue. Its eigenvector has positive entries.Pro of. The pro of is quite geometric and intuitive. Look at the sphere x21+. . .+x2n= 1 and intersectit with the space {x1≥ 0 , . . . , xn≥ 0 } which is a quadrant for n = 2 a nd octant for n = 3. Thisgives a closed, bounded set X. The matrix A defines a map T (v) = Av/|Av| on X because theentries of the matrix are nonnegative. Because they are positive, T X is contained in the interiorof X. This map is a contraction, there exists 0 < k < 1 such that d(T x, T y) ≤ kd(x, y) where dis the g eodesic sphere distance. Such a map has a unique fixed point v by Banach’s fixed pointtheorem. This is the eigenvector Av = λv we were looking for. We have seen now that on X,there is only one eigenvector. Every other eigenvector Aw = µw must have a coordinate entrywhich is negative. Write |w| for the vector with coordinates |wj|. The computation|µ||w|i= |µ wi| = |XjAijwj| ≤Xj|Aij||wj| =XjAij|wj| = ( A|w|)ishows that |µ|L ≤ λL because ( A|w|) is a vector with length smaller than λL, where L is thelength of w. From |µ|L ≤ λL with nonzero L we get |µ| ≤ λ. The first ”≤” which appears in thedisplayed formula is however an inequality for some i if one of the coordinate entries is negative.Having established |µ| < λ the proof is finished.Remark. The theorem generalizes to situations considered in chaos theory, where productsof random matrices are considered which all have the same distribution but which do not needto be independent. Given such a sequence of random matrices Ak, define Sn= An· An−1· · · A1.This is a non commutative analogue of the random walk Sn= X1+ ... + Xnfor usual randomvariables. But it is much more intricate because matrices do not commute. Laws of large numbersare now more subtle.Application: ChaosThe Lyapunov exponent of a random sequence of matrices is defined aslimn→∞12nlog λ(STnSn) ,where λ(B) is the maximal eigenvalue of the symmetric matrix STnSn.Here is a prototype result in Chaos theory due to Anosov for which the proof o f Perron-Frobeniuscan be modified using different contractions. It can be seen as an example of a noncommutativelaw of large numbers:If Akis a sequence o f identically distributed random positive matrices of determinant1, then the Lyapunov exponent is positive.4 Let Akbe either"2 11 1#or"3 21 1#with probability 1/2. Since the matrices do notcommute, we can not determine the long term behavior of Snso easily and laws of la r genumbers do not apply. The Perron-Frob enius generalization above however shows that still,Sngrows exponentially fast.Positive Lyapunov exponent is also called sensitive dependence on initial conditions forthe system or simply dubbed ”chaos”. Nearby trajectories will deviate exponentially. EdwardLorentz, who studied about 50 years ago models of the complicated equations which govern ourweather stated this in a poetic way in 1972:The flap of a butterfly’s wing in Bra zil can set off a tornado in Texas.Unfortunately, Mathematics is quite weak still to mathematically prove positive Lyapunov ex-ponents if the system does not a priori feature positive matrices. There are cases which can besettled quite easily. For example, if the matrices Akare IID random matrices of determinant 1and eigenvalues 1 have not full probability, then the Lyapunov exponent is positive due to workof Fuerstenberg and others. In real systems, like for the motion of our solar system or particlesin a box, positive Lyapunov exponents is measured but can not be proven yet. Even for simpletoy systems like Sn= dTn, where dT is the Jacobean of a map T like T (x, y) = (2x − c sin(x), y)and Tnis the n’th iterate, things are unsettled. One measures λ ≥ log(c/2) but is unable toprove it yet. For our real weather system, where the Navier stokes equations apply, one is evenmore helpless. One does not even know whether trajectories exist for all times. This existenceproblem looks like an esoteric ontological question if it were not for the fact that a one milliondollar bounty is offered for its solution.Application: PagerankA set of nodes with connections is a graph. Any network can be described by a graph. The linkstructure of t he web forms a graph, where the individual websites are the nodes and if there is anarrow from site aito site ajif ailinks to aj. The adjacency matrix A of this gr aph is called theweb graph. If there are n sites, then the adjacency matrix is a n × n matrix with entries Aij= 1if there exists a link from ajto ai. If we divide each column by the number of 1 in that column,we obtain a Markov matrix A which is called the normalized web matrix. Define the