Vladimir Protasov (Moscow State University) Perron-Frobenius theory for matrix semigroups

NO: The matrices may have a common invariant subspace

Example: 321 Question 1. How to characterize all families that have positive products ? Question 2. Is it possible to decide the existence of a positive product within polynomial time ?

The case of one matrix (m=1) If a family {A} possesses a positive product, then some power of A is positive. Definition 1. A matrix is called primitive if it has a strictly positive power. Primitive matrices share important spectral and dynamical properties with positive matrices and have been studied extensively.

Perron-Frobenius theorem (1912) A matrix A is not primitive if it is either reducible or one of the following equivalent conditions is satisfied:

Can these results be generalized to families of m matrices or to multiplicative matrix semigroups ? One of the ways – strongly primitive families.

Strongly primitive families have been studied in many works. Applications: inhomogeneous Markov chains, products of random matrices, probabilistic automata, weak ergodicity in mathematical demography. There is no generalization of Perron -Frobenius theory to strongly primitive families The algorithmic complexity of deciding the strong primitivity of a matrix family is unclear. Most likely, this is not polynomial. Let N be the least integer such that all products of length N are positive. There are families of d x d – matrices, for which N = (Cohen, Sellers, 1982; Wu, Zhu, 2015) (compare with N = for one matrix)

Another generalization: the concept of primitive families. Definition 3. A family of matrices is called primitive if there exists at least one positive product. Justification. If the matrices of the family have neither zero columns no zero rows, then almost all long products are positive.

Question 1. How to characterize primitive families ? Question 2. Is it possible to decide the existence of a positive product within polynomial time ? Can the Perron-Frobenius theory be somehow generalized to primitive families ? The answers to both these questions are affirmative. (under some mild assumptions on matrices)

The main results (conjectured in 2010)

Proofs of Theorem 1 (2012) P., Voynov. By applying geometry of affine maps of convex polyhedra. (2013) Alpin, Alpina. Combinatorial proof. (2014) Blondel, Jungers, Olshevsky. Combinatorial proof. (2015) P., Voynov. By applying functional difference equations. Call for purely combinatorial proofs

What about the minimal length N of the positive product ?

Another generalization: m-primitive families Fornasini, Valcher (1997), Olesky, Shader, van den Driessche (2002), etc. The family is m-primitive if it has at least one positive Hurwitz product

Our approach can be extended to m-primitive families. Applications for graphs and for multivariate (2D, 3D, etc.) Markov chains. The complexity of recognition of m-primitive families was unclear. There is a criterion, which is highly non-polynomial. The proof is algebraic, it uses the theory of abelian groups

Applications of primitivity of matrix families inhomogeneous Markov chains products of random matrices, Lyapunov exponents, probabilistic automata, refinement functional equations mathematical ecology (succession models for plants) Products of random matrices, Lyapunov exponents Every choice is independent with equal probabilities 1/m (the simplest model)

This result was significantly strengthened by V.Oseledec (multiplicative ergodic theorem, 1968) The problem of computing the Lyapunov exponent is algorithmically undecidable (Blondel, Tsitsiclis, 2000)

In case of nonnegative matrices there are good results on both problems. 1) An analogue of the central limit theorem for matrices (Watkins (1986), Hennion (1997), Ishitani (1997)) 2) Efficient methods for estimating and for computing the Lyapunov exponent (Key (1990), Gharavia, Anantharam (2005), Pollicott (2010), Jungers, P. (2011)). All those results hold only for primitive families. The existence of at least one positive product is always assumed in the literature ``to avoid pathological cases ’’ Our Theorems 1 and 2 extend all those results to general families of nonnegative matrices.

Refinement equation is a difference functional equation with the contraction of an argument is a sequence of complex numbers sutisfying some constraints. This is a usual difference equation, but with the double contraction of the argument Refinement equations with nonnegative coefficients Applications: wavelets theory, approximation theory, subdivision algorithms, power random series, combinatorial number theory.

How to check if in case all the coefficients are nonnegative ? I.Daubechies, D.Lagarias, 1991 A.Cavaretta, W.Dahmen, C.Micchelli, 1991 C.Heil, D.Strang, 1994 R.Q.Jia, 1995, K.S.Lau, J.Wang, 1995 Y.Wang, 1996 Example. How to check the existence of a compactly supported solution ?

Conclusions In particular, to construct an efficient algorithm of computing the Lyapunov exponents of nonnegative matrices. Thus, if a family of matrices is not primitive, then all its matrices constitute permutations of the canonical partition. The canonical partition can be found by a fast algorithm. This allows us to extend many results on Lyapunov exponents to general families of nonnegative matrices. Thank you! Other applications: functional equations, succession models in mathematical ecology, etc.