Words and Automata, Lecture 1 Dominique Perrin 18 octobre 2012 Dominique Perrin Words and Automata, Lecture 1
Outline of the minicourse Positive matrices Examples Irreducible and primitive matrices Perron Frobenius Theorem Probability distributions on words Information sources Entropy Distribution of maximal entropy Noncommutative stochastic matrices Dominique Perrin Words and Automata, Lecture 1
Example 1 : the adjacency matrix of an automaton a b 1 2 a The matrix � 1 � 1 M = 1 0 is the adjacency matrix of the automaton. The coefficient ( i , j ) of M n is the number of paths of length n from i to j . Here � f n � f n − 1 M n = f n − 1 f n − 2 with f − 1 = 0, f 0 = 1 and f n +1 = f n + f n − 1 . Dominique Perrin Words and Automata, Lecture 1
Example 2 : a Markov chain 1 / 3 2 / 3 1 2 1 The matrix � 1 2 � 3 3 M = 1 0 is the transition matrix of the Markov chain. The vector � 3 2 � x = is the stationnary distribution of the chain. 5 5 The matrix M is called a stochastic matrix : it is such that the sum of the coefficients on each row is 1. Dominique Perrin Words and Automata, Lecture 1
Google’s ranking Let Q be the set of web pages (many...). Let G be the graph with edges the ( p , q ) such that page p points to page q . Let M be the adjacency matrix of G . The eigenvector v of M for a value ρ such that vM = ρ M gives the Google rank of the page p . Dominique Perrin Words and Automata, Lecture 1
Nonnegative matrices Let Q be a set of indices. For two Q -vectors v , w with real coordinates, one writes v ≤ w if v q ≤ w q for all q ∈ Q , and v < w if v q < w q for all q ∈ Q . A vector v is said to be nonnegative (resp. positive) if v ≥ 0 (resp. v > 0). In the same way, for two Q × Q -matrices M , N with real coefficients, one writes M ≤ N when M p , q ≤ N p , q for all p , q ∈ Q , and M < N when M p , q < N p , q for all p , q ∈ Q . The Q × Q -matrix M is said to be nonnegative (resp. positive ) if M ≥ 0 (resp. M > 0). We shall use often the elementary fact that if M > 0 and v ≥ 0 with v � = 0, then Mv > 0. Dominique Perrin Words and Automata, Lecture 1
Irreducible matrices A nonnegative matrix M is said to be irreducible if for all indices p , q > 0, where M k denotes p , q , there is an integer k such that M k the k –th power of M . Otherwise M is said to be reducible. Dominique Perrin Words and Automata, Lecture 1
Proposition A nonnegative matrix M is irreducible if and only if ( I + M ) n > 0 where n is the dimension of M. Proposition A nonnegative matrix M is reducible if and only if there is a reordering of the indices such that M is block triangular, i.e. of the form � U � V M = (1) 0 W with U , W of dimension > 0 . Dominique Perrin Words and Automata, Lecture 1
A nonnegative matrix M is called primitive if there is an integer k such that M k > 0. A primitive matrix is irreducible but the converse is not true. A nonnegative matrix M is called aperiodic if the greatest common divisor of the integers k such that M k i , i > 0 for some i is equal to 1 (including the case where the set of integers k is empty). Dominique Perrin Words and Automata, Lecture 1
Proposition A nonnegative matrix is primitive if and only if it is irreducible and aperiodic. Indeed, if M is primitive, it is clearly irreducible and aperiodic. Conversely, if M is not irreducible, it is not primitive. Next, if it is irreducible but periodic, consider the grpah G such that there is an edge from i to j if and only if M ij > 0. Let p > 1 be the gcd of the lengths of the paths from 1 to 1. Let i be such that M 1 i > 0. Then all paths from 1 to i have length k ≡ 1 mod p . Thus M is not primitive. Dominique Perrin Words and Automata, Lecture 1
The Perron–Frobenius Theorem The following result is part of a theorem known as the Perron–Frobenius theorem. It says in particular that the spectral radius of a nonnegative matrix is an eigenvalue. Theorem (Perron–Frobenius) Any nonnegative matrix M has a real eigenvalue ρ M such that | λ | ≤ ρ M for any eigenvalue λ of M, and there corresponds to ρ M a nonnegative eigenvector v. If M is irreducible, there corresponds to ρ M a positive eigenvector v. Observe that the same result holds both for right and for left eigenvectors. Dominique Perrin Words and Automata, Lecture 1
Example Let � 1 � 1 M = . 1 0 The matrix M is nonnegative and irreducible. The eigenvalues of √ √ M are ϕ = (1 + 5) / 2 and ˆ ϕ = (1 − 5) / 2. The vector � ϕ � 1 � � x = is an eigenvector relative to ϕ . The vector v = is 1 1 an approximate eigenvector relative to r = 1 and Mv is an approximate eigenvector relative to r = 3 / 2. Dominique Perrin Words and Automata, Lecture 1
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