Binary Factorizations of the Matrix of All Ones Maguy Trefois Paul - PowerPoint PPT Presentation

Binary Factorizations of the Matrix of All Ones Maguy Trefois Paul Van Dooren Jean-Charles Delvenne Université catholique de Louvain, Belgium ILAS, June 2013 1

Motivation : the finite-time average consensus problem We have: • n communicating agents with an initial position • a communication topology At each time step: • each agent sends its current position to some other agents according to the communication pattern • with the received information, each agent changes its position The goal: after a finite time, all the agents meet at the average of their initial positions 2

Vector of initial positions: � x ( 0 ) Dynamics: x ( t + 1 ) = A t + 1 .� � x ( 0 ) The matrix A respects the communication topology:   0 ? 0 0 0 0 0 ?   A =   0 ? 0 0   ? 0 ? 0 3

x ( t + 1 ) = A t + 1 .� � x ( 0 ) The matrix A is a solution to the consensus if:   0 ? 0 0 ? 0 0 0   • A is of the form   0 ? 0 0   ? ? 0 0   1 . . . 1 . . . • After a finite time m , A m = 1 . . . 4 .   . . .   . . . 1 1 4

x ( t + 1 ) = A t + 1 .� � x ( 0 ) The matrix A is a solution to the consensus if:   0 ? 0 0 ? 0 0 0   • A is of the form   0 ? 0 0   ? ? 0 0   1 . . . 1 . . . • After a finite time m , A m = 1 . . . 4 .   . . .   . . . 1 1 Question : for which communication patterns is it possible to reach the consensus ? 4

We should study the solutions to the equation:   1 . . . 1 A m = 1 . . . . . . n .  ,   . . .  1 . . . 1 where A ∈ R n × n . 5

We should study the solutions to the equation:   1 . . . 1 A m = 1 . . . . . . n .  ,   . . .  1 . . . 1 where A ∈ R n × n . ⇒ difficult to tackle directly 5

We should study the solutions to the equation:   1 . . . 1 A m = 1 . . . . . . n .  ,   . . .  1 . . . 1 where A ∈ R n × n . ⇒ difficult to tackle directly Simpler problem: study the solutions to:   1 . . . 1 A m = . . . . . .  ,   . . .  1 . . . 1 where A ∈ { 0 , 1 } n × n . 5

Outline Factorization problem The De Bruijn matrices Factorizations into commuting factors General form of a root of I n with minimum rank A root class of I n Conclusion 6

Factorization problem The De Bruijn matrices Factorizations into commuting factors General form of a root of I n with minimum rank A root class of I n Conclusion 7

We are looking for the solutions to m � A i = A 1 A 2 ... A m = I n , i = 1 where - I n is the n × n matrix with all ones - each factor A i is an n × n binary matrix. 8

We are looking for the solutions to m � A i = A 1 A 2 ... A m = I n , i = 1 where - I n is the n × n matrix with all ones - each factor A i is an n × n binary matrix. In particular, we are investigating the solutions to: A m = I n , where A is a binary matrix. 8

Factorization problem The De Bruijn matrices Factorizations into commuting factors General form of a root of I n with minimum rank A root class of I n Conclusion 9

Lemma If A ∈ { 0 , 1 } n × n is such that A m = I n , then - A is p-regular, i.e A . 1 = p . 1 and 1 T . A = p . 1 T - n = p m 10

Lemma If A ∈ { 0 , 1 } n × n is such that A m = I n , then - A is p-regular, i.e A . 1 = p . 1 and 1 T . A = p . 1 T - n = p m Definition The De Bruijn matrix of order p and dimension n is a matrix of the form: D ( p , n ) := 1 p ⊗ I n / p ⊗ 1 T p , where - I n / p is the identity matrix of dimension n / p - 1 p is the p × 1 vector with all ones - ⊗ denotes the Kronecker product Moreover, it is imposed that n = p m , for some integer m. 10

1 1 0 0 0 0 0 0   0 0 1 1 0 0 0 0     0 0 0 0 1 1 0 0     0 0 0 0 0 0 1 1   D ( 2 , 8 ) =   1 1 0 0 0 0 0 0     0 0 1 1 0 0 0 0     0 0 0 0 1 1 0 0   0 0 0 0 0 0 1 1 11

1 1 0 0 0 0 0 0   0 0 1 1 0 0 0 0     0 0 0 0 1 1 0 0     0 0 0 0 0 0 1 1   D ( 2 , 8 ) =   1 1 0 0 0 0 0 0     0 0 1 1 0 0 0 0     0 0 0 0 1 1 0 0   0 0 0 0 0 0 1 1 Proposition The De Bruijn matrix D ( p , n ) with n = p m is such that D ( p , n ) m = I n . 11

1 1 0 0 0 0 0 0   0 0 1 1 0 0 0 0     0 0 0 0 1 1 0 0     0 0 0 0 0 0 1 1   D ( 2 , 8 ) =   1 1 0 0 0 0 0 0     0 0 1 1 0 0 0 0     0 0 0 0 1 1 0 0   0 0 0 0 0 0 1 1 Proposition The De Bruijn matrix D ( p , n ) with n = p m is such that D ( p , n ) m = I n . Question : Can we characterize all the roots from the De Bruijn matrices ? 11

Factorization problem The De Bruijn matrices Factorizations into commuting factors General form of a root of I n with minimum rank A root class of I n Conclusion 12

Factorization into commuting factors : Looking for the solutions to: AB = BA = I n , where - A and B are binary matrices - A is p -regular - B is l -regular 13

Factorization problem: AB = BA = I n , where A is p -regular and B is l -regular. Theorem If A and B are commuting factors, then • p . l = n • rank ( A ) ≥ n / p and rank ( B ) ≥ n / l • if rank ( A ) = n / p (resp. rank ( B ) = n / l), then there exist permutation matrices P 1 , P 2 such that P 1 AP T ( resp. P 2 BP T 2 = D ( p , n ) 1 = D ( l , n )) . 14

Question : Is it possible that rank ( A ) > n / p ?  1 0 1 0   1 1 0 0  0 1 0 1 1 1 0 0     A = B =     1 0 0 1 0 0 1 1     0 1 1 0 0 0 1 1 - A and B are 2-regular - AB = BA = I 4 - BUT, rank ( A ) = 3 > 4 / 2 15

Question : Can we choose P 1 = P 2 ?  0 1 0 1 0 1   1 1 0 0 0 0  1 0 1 0 1 0 0 0 1 1 0 0         0 1 0 1 0 1 0 0 0 0 1 1     A = B =     1 0 1 0 1 0 1 1 0 0 0 0         0 1 0 1 0 1 0 0 1 1 0 0     1 0 1 0 1 0 0 0 0 0 1 1 - A is 3-regular, B is 2-regular and AB = BA = I 6 - rank ( A ) = 6 / 3, rank ( B ) = 6 / 2 - BUT, A is not isomorphic to D ( 3 , 6 ) since  3 0 3 0 3 0   2 2 2 1 1 1  0 3 0 3 0 3 1 1 1 2 2 2         3 0 3 0 3 0 2 2 2 1 1 1 A 2 = , D ( 3 , 6 ) 2 =         0 3 0 3 0 3 1 1 1 2 2 2         3 0 3 0 3 0 2 2 2 1 1 1     0 3 0 3 0 3 1 1 1 2 2 2 16

Corollary Let A be a binary matrix satisfying A m = I n . Then, - A is p-regular - if rank ( A ) = n / p, then there are permutation matrices P 1 , P 2 such that P 1 AP T 2 = D ( p , n ) . As previously, - A may have a rank greater than n / p - A may not be isomorphic to D ( p , n ) even though rank ( A ) = n / p 17

Factorization problem The De Bruijn matrices Factorizations into commuting factors General form of a root of I n with minimum rank A root class of I n Conclusion 18

Theorem Let A ∈ { 0 , 1 } n × n such that A m = I n , A is p-regular and p m = n. If rank ( A ) = n / p, then A is isomorphic to a matrix P 1 D ( p , n ) , where P 1 = diag ( Q 1 , ..., Q p ) with each Q i ∈ { 0 , 1 } n / p × n / p is a permutation matrix. 19

Theorem Let A ∈ { 0 , 1 } n × n such that A m = I n , A is p-regular and p m = n. If rank ( A ) = n / p, then A is isomorphic to a matrix P 1 D ( p , n ) , where P 1 = diag ( Q 1 , ..., Q p ) with each Q i ∈ { 0 , 1 } n / p × n / p is a permutation matrix. Not all the matrices of that form are solutions. Indeed, consider 1 1 0 0 0 0 0 0   0 0 1 1 0 0 0 0     0 0 0 0 1 1 0 0     0 0 0 0 0 0 1 1   A =   1 1 0 0 0 0 0 0     0 0 0 0 0 0 1 1     0 0 0 0 1 1 0 0   0 0 1 1 0 0 0 0 19

Factorization problem The De Bruijn matrices Factorizations into commuting factors General form of a root of I n with minimum rank A root class of I n Conclusion 20

A 2-circulant matrix:  0 1 0 1 0  1 0 0 1 0     1 0 1 0 0     0 0 1 0 1   0 1 0 0 1 Theorem (Wu, 2002) Let A ∈ { 0 , 1 } n × n be g-circulant and such that A m = I n . If - g m ≡ 0 mod n - A is p-regular, then A is isomorphic to D ( p , n ) . 21

Definition A nice permutation matrix is built as follows: start with a p × p permutation matrix. Then, replace all the zeros by a zero p × p matrix and each one by a p × p permutation matrix. Repeat this m times. You obtain a permutation matrix of dimension p m . 22