Vladimir Protasov (Moscow State University)

Perron-Frobenius theory for matrix semigroups

1Let A , , A be nonnegative matrices. m d d

NO: The matrices may have a common invariant subspace

1

, 1,...,

Every product of these matrices has the

A0

0 same form A A

there are no positive products.

k

i i

ii

d d

B Ci m

D

1Let us assume that the family A , , A is ir

Does a positive prod

re

uc

ducib

t exist ?

le. m

1 A , , A are all permutation matri

0 1 0

0 0 1

Exampl ces.

, 1,

e:

1 0

,

0

m

i mA i

!No

Do they have a product, which is strictly positive ?

1

1

2

1 A , , A can be equivalent to permutation matrices.

, 1, ,

All products are also permutation

0

matrices there are no positi

0

0 0

0 0

ve products.k

m

d

i

d

d

i m

A A

a

aA

a

1

1,..,

, , constitute permutations of the same partition of the set: {1,..., }.

Let = be a partition of the set to disjoint sets.

The

More general:

matrix acts as a permu

m

jj r

j

A A d

r

A

1tation of the sets ,...,

there are no positive products.

r

321 4 5 6 7 81,..,4

= jj

:iA

Example: 321 1 2

0 * * * 0 0

A = * 0 0 ; A = 0 * *

* 0 0 0 * *

1 2The family {A , A } has no positive produ cts.

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 ?

1 do nonnegative matrices A , , A have a zero prod

This problem is NP-complete already for

An ''opposite''

2 (Blondel

uct ?

, Tsi

p

tsiklis,

roblem:

1997)

md

m

d

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.

* 0 * 0

* 0 0 0A =

0 * 0 *

* 0 * 0

N

* * * *

* * * *A =

* * * *

* * * *

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

there are 2 eigenvalues of A, counting with multiplicities, equal by modulo

to its spectral radius

1)

( ).

r

A

The number is the . of the matrix r imprimitivity index A

2

They are all different and equal to = , 0,... , 1. ik

rk e k r

All cycles of the graph of the matr (Roman ix hovsky, 1933 ave g.c.d.2) =) . A r

1

1

0 0

0 0 On the basis corresponding to that partition 4 has the) form:

0 0

r

r

B

BA A

B

1There is a partition of the set {1,..., } into sets ,..., ,

on which acts as a cycli

3)

c permuta

tion.

rd r

A

Can these results be generalized to families of m matrices or to multiplicative matrix semigroups ?

2

(H.Wielandt, 1950),

Proofs: Rosenblatt, Hollad

If a matrix is primitive, then 0 , whenever

ay and Varga, Ptak, and Dionisio, Senet

( -1) 1.

The primitivity of a matri

a .

NA A N d

x can be decided within polynomial time.

One of the ways – strongly primitive families.

1

1A family { ,..., } is called if there is N

such that every product

Defi

of length is positive

nition 2

.

.

k

m

d d

A A strongly primitive

A A k N

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)

d 2 2

2( -1) 1d

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.

1

1 1

Let ( ) be the number of positive products among all products of length .

( )Then 1 as .

If each matrix in the product is chosen from the set { , ,

k

k k

kd d

k

d d d

g k m A A k

g kk

mA A A A

}

independently with probability 1/ , then the product is positive with probability 1.

mA

p m

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 ?

1

1,..,

Is it true that if the family is not primitive, then all , ,

constitute permutations of the same partition of the set: {1,..., }, = ?

m

jj r

A A

m

The answers to both these questions are affirmative.

(under some mild assumptions on matrices)

The main results

1

If a family of matrices satisfies (a) and (b), then it either possesses

a positive product, or there is a partition of the set

[P., Voynov, 2012].

{1,..., } into 2 sets ,...

Theorem 1

, rd r ,

on which every matrix acts as a permutation.jA

1

1( )

Let us have a family { ,..., } of nonnegative matrices. We make the following two assumptions:

The family is irreducible, i.e., the matrices ,..., do not share a common invariant coord

m

m

A A

Aa A inate plane.

All matrices of the family have neither zero rows nor zero col( ) um ns.b

d

1

2

This means that there is a permutation of the basis of , after which every matrix

gets the block form:

0 0

0 0

Remar

0 0

k 1. j

j

r

A

B

BA

B

dR

The permutations are not necessaril (In contraR sy t cycl to temark he P-ic F t h2. eo! rem)

(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

Among all the partitions , there is a unique partition

with the maximal number

Theorem

of parts .

2.

jj

r

In particular, the family is almost primitive 1.

Hence, that algorithm also decides the existence of a positi

Remark

ve prod t

.

3.

uc

r

2

3/ 2

2.376

Rem The size of the instance of the algorithm is .

So, its complexity is less than 2 .

The computing of product of two -mat

ark

rices takes operations

(in

practice i

4.

t i

N md

N

d d C d

3s operations). So, the complexity estimate of deciding primitivity

can hardly be improved significantly.

C d

is the imprimitivity index of the family r

3This ``canonical'' patition can be found by an algorithm spending 2 arithmetic operations.m d

The is also related to the multiplicity

of the largest by modulo eigenvalues of matrices (as in the P-F theo

Remark 5.

rem).

imprimitivity index r

For a family of stochastic matrices the imprimitivity index equals to

the minimal total multiplicity of the largest by modulo eigenvalues in the matrix

semigroup generated

Theorem 3.

by this fam

r

ily.

2

However, now the leading eigenvalues are not necessarily the roots of unity

= , 0,... , 1.ik

rk e k r

Both assumptions (a) and (b) are essential. For (a) this is obvious, for (b) there aRema re erk 5. xamp les.

1

1Recall that all these results hold under the assumptions that the family { ,..., } satisfies

The family is irreducible, i.e., the matrices ,..., do not share a common invariant coord( in a) m

m

A A

A

a

A

te plane.

All matrices of the family have neither zero rows nor zero col( ) um ns.b

1 2 3 4

1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0

0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0A ; A ;

The set of four matrices

is not primitive, h

A ; A0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1

0 0 0 0 0 0 0 0 1 0 0

ow

Example 1.

ever, it does

1 1 0 1 0

n

ot have a partition.

condition (b) is violated (all the matrices have zeroRea roson: ws).

Without condition (b), the problem

(Blondel, Jungers, Olshevsky, 2014

of deciding primitivity is NP-h

)

ard.

What about the minimal length N of the positive product ?

3 2If the positive product exists, then 1

(Voynov, 20 O( ) 2

13) N d d

2 3 2

Find a sharp

estimate for N. By now

Problem.

1( -

we know only that

1) 1 ( ) 3

d N d o d

If the length of the shortest sincronizing word

in a deterministic automata is

(Blondel, Jungers, O

( ), then

lshevsky,

2 ( ) - 1

2014)

f d N f d d 3

(P in, ( ) 6

1983) d d

f d

3 2This gives an estimate1

N ( )3

: d o d

Another generalization: m-primitive familiesFornasini, Valcher (1997), Olesky, Shader, van den Driessche (2002), etc.

1 1

1 1

2 2

A family A , , A is called m-primitive if there are numbers , ... ,

such that the sum of all products of length with matrices ,

matrices , ... ,

Definition 4.

m

m m

i

m

k k

N k k A

k A k

atrices , is positive. mA

1( ,..., )1

1 1 2 2

Hurwitz product corresponding to a ''color vector'' ( ,..., )

is the sum of all products with matrices , matrices , ... , matrices

Defini

.

tion 5

. mk km

m m

A k k

k A k A k A

(2,0,1) 2 21 3 1 3 1 3 1 = Example. A A A A A A A A

1 1

1

( ,..., )1 1 1

...

( ... ) .... m m

m

k k kkNm m m

k k N

t A t A A t t

The family is m-primitive if it has at least one positive Hurwitz product

Our approach can be extended to m-primitive families.

If all matrices of the family have no zero columns no zero rows ,

then it is either m-primitive, or there is a partition of the set

[P.Theorem 5

{1,...

, 2

, } into 2 sets

0 (!

3] )1 jA or

d r 1 ,..., ,

on which all matrices act as permutations.(!)

r

jA commuting

3 2 2

Under the assumptions of Theorem 5, the m-primitivity of a set of matrices

1is decidable within polynomial time. The complexity is 2 arithmetic operations.

2

Theorem 6.

md m d

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.

1 F

(Olesky, Shader, van den Driessche (200

or the minimal length N of products we ha

)

ve

2

)

mN C d

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 equationsmathematical ecology (succession models for plants)

Products of random matrices, Lyapunov exponents

1

1

1

Let { ,..., } be a given family of arbitrary matrices.

Consider an infinite product , where each matrix

is chosen from the family { , , } independently with probabili

ty k k

m

d d d

m

A A

A A A

A A

1/ .p m

1 1kd dA A

kd

A

1A

2A

mA

Every choice is independent with equal probabilities 1/m (the simplest model)

1

1/

The ``mean rate of growth'', i.e., the value converges

to a constant with probability 1

This is a matrix analo

(Furstenberg,

gue of the la

Kesten, 19

w of large

60).

numbers.

T

va

he

k

k

d dA A

1

2 1

2

lue log is called the of the family { , , }.

1 is the ``spectral radius'' of the family, and

Lyapunov expone

= lim

lo

t

g

n

k

m

d dk

A A

E A Ak

This result was significantly strengthened by V.Oseledec (multiplicative ergodic theorem, 1968)

1

1/

The main problems:

to characterize the convergence

to compute or estimate for a given matrix family.

k

k

d dA A

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

0 , , Nc c is a sequence of complex numbers sutisfying some constraints.

( )x0 (2 )c x

1 (2 1)c x

.......

(2 )Nc x N

0

( ) (2 ) ,N

kk

x c x k

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 ? ( )pL R

I.Daubechies, D.Lagarias, 1991A.Cavaretta, W.Dahmen, C.Micchelli, 1991

C.Heil, D.Strang, 19940 1the family of matrices { , } is strongly pr( ) imitive T TC R

R.Q.Jia, 1995, K.S.Lau, J.Wang, 1995Y.Wang, 1996

20 1 1, are matric , )s (e i j k j k iT T N N T c

0

2 1 00

4 3 2 1

4 3

0 0 0

0

0 0

c

c c cT

c c c c

c c

1 0

3 2 1 01

4 3 2

4

0 0

0

0 0 0

c c

c c c cT

c c c

c

0 1 2 3 4 4, , , , ,N c c c c cExample.

How to check the existence of a compactly supported solution ?

0 1the family { , } is primitive ( )p T TL R

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.