Zero forcing, minimum rank, and applications to control of … · 2012. 7. 18. · Zero forcing & quantum control Leslie Hogben Introduction Zero forcing Matrices and Graphs Lie Algebras

Zero forcing &quantum control

Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results

Zero forcing results

Zero forcing, minimum rank, andapplications to control of quantum systems

Leslie Hogben

Iowa State University andAmerican Institute of Mathematics

Joint work with Daniel Burgarth, Domenico D’Alessandro, SimoneSeverini, Michael Young

SIAM Annual ConferenceMinneapolis, MN, July 12, 2012


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


IntroductionZero forcingMatrices and GraphsLie AlgebrasControl

Results on controllabilityMatrix resultsZero forcing results


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero Forcing Number

Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.

Example


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero Forcing Number


Example


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero Forcing Number


Example


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero Forcing Number


Example


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero Forcing Number


Example


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero Forcing Number


Example


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero Forcing Number


Example


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero Forcing Number


Example


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero Forcing Number


Example


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero Forcing Number


Example


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero Forcing Number


Example


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero Forcing Number


Example


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


I A subset Z ⊆ VG defines a coloring by coloring allvertices in Z black and all the vertices not in Z white.

I Color change rule: If u is a black vertex of G , andexactly one neighbor v of u is white, then change thecolor of v to black.

I A zero forcing set for a graph G is a subset of verticesZ such that if initially the vertices in Z are coloredblack and the remaining vertices are colored white, thethe result of applying the color change rule until nomore changes result is that all vertices are black.

I The zero forcing number Z(G ) is the minimum of |Z |over all zero forcing sets Z ⊆ V (G ).

Zero forcing is also called graph infection.

Zero forcing is used in the study of minimum rank/maximumnullity problems and in control of quantum systems.


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Matrices and Graphs

Hn(R) denotes the real vector space of real symmetricmatrices.

The graph of A ∈ Hn(R) is G(A) = (V ,E ) where

I V = {1, ..., n},I E = {ij : aij 6= 0 and i 6= j}.I Diagonal of A is ignored.

Example: G(A)

A =

2 −1 3 5−1 0 0 0

3 0 −3 05 0 0 0

1 2

34

G(AG ) = G and G(LG ) = G , where AG is the adjacencymatrix and LG is the Laplacian matrix.


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Maximum nullity and minimum rank

The maximum nullity of graph G is

M(G ) = max{nullA : A ∈ Hn(R),G(A) = G}.

The minimum rank of graph G is

mr(G ) = min{rankA : A ∈ Hn(R),G(A) = G}.

I M(G ) + mr(G ) = |G |.

Theorem (AIM08)

For any graph G, M(G ) ≤ Z (G ).


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Lie algebras

For A1, . . . ,Ak ∈ Cn×n,

〈A1, . . . ,Ak〉[·,·]

is the real Lie algebra generated by A1, . . . ,Ak underaddition, real scalar multiplication, and the commutatoroperation.

gl(n,R) is the Lie algebra of all n × n real matrices(i.e., gl(n,R) = Rn×n).

u(n) is the Lie algebra of n × n skew-Hermitian matrices(over R).

su(n) is the Lie algebra of n × n skew-Hermitian matriceswith trace zero (over R).


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


For A1, . . . ,Ak ∈ Hn(R),

〈A1, . . . ,Ak〉[·,·] = gl(n,R) ⇐⇒ 〈iA1, . . . , iAk〉[·,·] = u(n).

For A ∈ Hn(R) and Z = {z1, . . . , zs} ⊂ Rn, the real Liealgebra generated by A and Z is

L(A,Z ) := 〈A, z1z1T , . . . , zszsT 〉[·,·].


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Linear control

A ∈ Rn×n, bj ∈ Rn, j = 1, 2, . . . , s, B = [b1 . . . bs ] ∈ Rn×s .

Linear system

x = Ax +s∑

j=1

bjuj ,

Control matrix

C(A,B) := [B,AB, . . . ,An−1B]

Classical controllability conditionThe linear system is controllable if and only ifrank C(A,B) = n.


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Walk matrix

A ∈ Rn×n, bj ∈ Rn, j = 1, 2, . . . , s,B = [b1 . . . bs ] ∈ Rn×s , Z = {b1, . . . ,bs}.

Walk matrix

W (A,b1) := [b1,Ab1, . . . ,An−1b1],

Walk matrix (extended)

W (A,Z ) := [b1,Ab1, . . . ,An−1b1, . . . ,bs ,Abs , . . . ,A

n−1bs ]

W (A,B) := W (A,Z )


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


A ∈ Rn×n, bj ∈ Rn, j = 1, 2, . . . , s, B = [b1 . . . bs ] ∈ Rn×s .

x = Ax +s∑

j=1

bjuj

Walk matrix (extended)

W (A,B) := [b1,Ab1, . . . ,An−1b1, . . . ,bs ,Abs , . . . ,A

n−1bs ],

Control matrix

C(A,B) := [B,AB, . . . ,An−1B]

The linear system is controllable if and only ifrank W (A,B) = n.


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Quantum control

For a finite dimensional closed quantum mechanical system:

Schrodinger equation

id

dt|ψ〉 = H(u)|ψ〉,

where |ψ〉 ∈ Cn is the quantum state, the Hamiltonianmatrix H = H(u) is Hermitian and depends on a controlu = u(t).


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


For a system linear in the state |ψ〉, the solution of theSchrodinger equation is |ψ(t)〉 = X (t)|ψ(0)〉 whereX = X (t) is the solution of

Schrodinger matrix equation

i X = H(u)X

with initial condition X (0) = In.

The solution X (t) is unitary at every time t.


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Quantum control

Schrodinger matrix equation

i X = H(u)X

Lie algebra rank conditionThe Lie algebra generated by the matrices iH(u) is u(n) orsu(n) (as u varies in the set of admissible values for thecontrol).

Quantum controllability conditionThe system described by the Schrodinger matrix equation iscompletely controllable if and only if Lie algebra rankcondition is satisfied.


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Matrix resultsQuantum control and walk matrices

Let A ∈ Hn(R), z ∈ Rn.

Theorem (Godsil Severini 2010)

If rankW (A, {z}) = n, then L(A, {z}) = gl(n,R).

TheoremIf L(A, {z}) = gl(n,R), then rankW (A, {z}) = n.


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Quantum control and (extended) walk matrices

Let A ∈ Hn(R) such that G(A) is connected and all thenonzero off-diagonal elements of A have the same sign.Let S ⊆ {1, . . . , n} and Z = {ej : j ∈ S}.

Theoremrank W (A,Z ) = n if and only if L(A,Z ) = gl(n,R).

Corollary

rank W (A, {ej : j ∈ S}) = n, i.e., the linear systemx = Ax +

∑sj=1 ejuj is controllable, if and only if

〈iA, {iejej T : j ∈ S}〉[·,·] = u(n), i.e., the quantum system

associated with the Hamiltonians iA and iejeTj , j ∈ S, is

controllable.


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


The hypothesis G(A) is connected is necessary.

Example Let A =

[A1 00 A2

]with Ai being ni × ni .

Let zi ∈ Rni , i = 1, 2 such that rank W (Ai , {zi}) = ni fori = 1, 2. Define z1 := [zT1 , 0

T ]T and z2 := [0T , zT1 ]T .

Then W (A, {z1, z2}) has rank n, but L(A, {z1, z2) containsonly block diagonal matrices.


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


The hypothesis all the nonzero off-diagonal elements of Ahave the same sign is necessary.

Example Let A =

0 1 0 11 0 −1 00 −1 0 11 0 1 0

, and Z = {e1, e3}.

Then rank W (A, {e1, e3}) = 4. However,dimL(A, {e1, e3}) ≤ 8 (by exhibiting an 8-dimensional Liesubalgebra of gl(4,R) containing A, e1eT1 , e3e

T3 ).

Since dim gl(4,R) = 16, L(A, {e1, e3}) 6= gl(4,R).


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Zero forcing and quantum control

Let A ∈ Hn(R) such that G(A) is connected and all thenonzero off-diagonal entries of A have the same sign. LetV := {1, 2, . . . , n} be the set of vertices for G(A), andZ ⊆ V be a zero forcing set of G(A).

TheoremL(A, {ejej T : j ∈ Z}) = gl(n,R).

Corollary

〈iA, {iejej T : j ∈ Z}〉[·,·] = u(n)and the corresponding quantum system is controllable.


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


The converse is false.

Example Consider the path on four vertices P4 with thevertices numbered in order. The set {e2} is not a zeroforcing set for P4. However,

W (AP4 , {e2}) =

0 1 0 21 0 2 00 1 0 30 0 1 0

and rank W (AP4 , {e2}) = 4, so L(AP4 , {e2}) = gl(n,R) byour previous results.


Leslie Hogben

Introduction

Zero forcing

Matrices and Graphs

Lie Algebras

Control

Results

Matrix results


Thank you!

Documents

Zero forcing, minimum rank, and applications to control of … · 2012. 7. 18. · Zero forcing & quantum control Leslie Hogben Introduction Zero forcing Matrices and Graphs Lie Algebras