introduction to combinatorial matrix theory

7/31/2019 Introduction to Combinatorial Matrix Theory

1/69


2/69

Combinatorial Matrix TheoryMinimum Rank Problems

Matrix Completion ProblemsReferences

Combinatorial Matrix Theory

ApplicationsMatricesGraph Terminology

Minimum Rank ProblemsMinimum Rank Problems for Simple GraphsMinimum Rank Problems for Simple TreesSimple GraphsSign PatternsMinimum Rank Problems for Tree Sign Patterns

Matrix Completion ProblemsDefinitionsGraph Theoretic TechniquesThe Positive (Semi)Definite Matrix Completion ProblemsThe (Strictly) Copositive Matrix Completion Problems

References


3/69





- Studies patterns of entries in a matrix rather than values.

- In some applications, only the sign of the entry (or whether itis nonzero) is known, not the numerical value.

- In other cases, some entries are missing.

- Uses graphs or digraphs to describe patterns.


4/69




Applications: Sign Patterns

- Study of sign patterns arose in economics and has applicationsto ecology.

- Behavior of supply and demand equilibrium or predator-prey

systems under perturbation can be predicted by a sign patternmatrix of partial derivatives whose signs are known.

- Sign stable patterns guarantee stability.

- Sign nonsingular (SNS) patterns guarantee invertibility andunique solution of a system.

- SNS patterns were first studied in the Dimer Problem instatistical mechanics involving the bonding of atoms in amolecule.


5/69




Applications: Matrix Completions

- Useful when some data unavailable but full matrix has certaintype.

- Used in seismology and geophysics (positive definite

matrices).

- Biomolecular modeling (Euclidean distance matrices).

- Image enhancement.


6/69




All (numerical) matrices discussed are real and symmetric, but the

ideas can be generalized to nonsymmetric matrices.Submatrices

- The principal submatrix A[] consists of the entries in rowsand columns in (i.e., delete rows and columns not in ).

- The principal submatrix obtained by deleting row and columnk is denoted A(k).

Example:

A = 3 2 0 0

2 1 1 10 1 1 30 1 3 1

A(2) = 3 0 00 1 30 3 1

A[{1,

2}] = 3 2

2 1


7/69




Eigenvalue Interlacing

Theorem

If A is a real symmetric matrix, then the eigenvalues of A(k)interlace the eigenvalues of A.

Example:

A =

3 2 0 02 1 1 10 1 1 3

0 1 3 1

A(2) =

3 0 00 1 3

0 3 1

(A) = {5, 3.56155,0.561553,2} (A(2)) = {4, 3,2}

5

4

3.56155

3

0.561553

2

2


8/69




Graph Terminology

- A simple graph G does not allow loops or multiple edges.- A graph G allows loops but does not allow multiple edges.

- The simple graph associated with G, denoted

G, is obtained

from G by suppressing loops.

Example:

G G


9/69




- A path is a sequence of distinct vertices v0, v1, v2, . . . , vk such

that for i = 1, . . . , k, vi1vi is an edge.- A cycle is a sequence of vertices v1, v2, . . . , vk, v1 such that

i = j vi = vj and vkv1 is an edge, vi1vi is an edgei = 2, . . . , k.

Example:A path and a cycle.

C bi i l M i Th


10/69




- A graph is connected if there is a path from any vertex to any

other vertex.- A graph is acyclic if it has no cycles.

- A graph T is a tree if T is connected and T is acyclic.

Example:

A tree.

C bi t i l M t i Th


11/69




Matrices and graphs

- We will associate matrices with graphs in several ways.

- With an n

n matrix we associate a graph having vertices

{1, . . . , n}.- The i,j entry will be associated with the edge (or non-edge)

between i and j.



12/69




Permutation Similarity and Vertex Numbering

- Applying a permutation similarity to a matrix does not changeits eigenvalues or its rank.

- Applying a permutation similarity to a symmetric matrix Acorresponds to renumbering the vertices of the graph of A

- Unlabeled graph diagrams can be used.



13/69




Block Diagonal Matrices and Graph Components

- If A = A1 A2 Ak, then the spectrum of A is theunion of the spectra of A1,A2, . . . , Ak andrank(A) =

ki=1 rank(Ai).

- Any symmetric matrix is permutation similar to a direct sumof irreducible matrices (block diagonal matrix).

- The irreducible summands of A correspond to the connectedcomponents of the graph of A.

- Only connected graphs are studied.

Combinatorial Matrix Theory Minimum Rank Problems for Simple Graphsf S


14/69



Minimum Rank Problems for Simple TreesSimple GraphsSign PatternsMinimum Rank Problems for Tree Sign Patterns

Let A be a symmetric n n matrix.- The simple graph G(A) of A is the graph having

- vertices 1, . . . , n- edge ij of G(A) if and only if i = j and aij = 0.

- The diagonal is ignored.Example: G(A)

A =

2 1 3 51 0 0 03 0 3 05 0 0 0

1 2

34

Combinatorial Matrix Theory Minimum Rank Problems for Simple GraphsMi i R k P bl f Si l T


15/69




Minimum Rank Problems for Simple Graphs

Let A be a symmetric n n matrix and G a simple graph.-

S(

G) = {A : A is a symmetric matrix and

G(A) =

G}.

- The minimum rank of G ismr(G) = min

AbS( bG){rank(A)}.

Combinatorial Matrix Theory Minimum Rank Problems for Simple GraphsMi i R k P bl s f Si l T s


16/69

yMinimum Rank Problems



Let A be a symmetric n

n matrix and G a simple graph.

- mult(A) denotes the multiplicity of eigenvalue of A.

- The maximum multiplicity of an eigenvalue in

S(

G) is

M(G) = maxAbS( bG)

{mult(A)}.

-

mr(

G) +

M(

G) = the number of vertices of

G.

Combinatorial Matrix Theory Minimum Rank Problems for Simple GraphsMinimum Rank Problems for Simple Trees


17/69

yMinimum Rank Problems



Let

G have n vertices.

- It is easy to obtain a matrix A S(G) with rank A = n 1(translate).

- It is easy to have all eigenvalues distinct (i.e., rank A n 1).- For a path,

mr(

Pn) = n 1.

Example: # is nonzero, is indefinite

A =

# 0 . . . 0 0# # . . . 0 00 # . . . 0 0...

..

.

..

. . . .

..

.

..

.0 0 0 . . . #0 0 0 . . . #


18/69

Combinatorial Matrix TheoryMi i R k P bl

Minimum Rank Problems for Simple GraphsMinimum Rank Problems for Simple Trees


19/69

Minimum Rank ProblemsMatrix Completion Problems

References


Algorithm (for computing , and thus M and mr)Let T be a simple tree.Set Q = (set of vertices to be deleted).Set H = {vertices of

T with degree 3} (candidates for deletion).

While H

=

:

1. Set TH = the unique component of T Q that contains anH-vertex.

2. Set W = {w H:at most 1 component of

TH

w is not H-free

}.

3. Q = Q W .4. H = H W .5. For each v H, ifdeg bTQv 2, remove v from H.

Combinatorial Matrix TheoryMi i R k P bl



20/69


References


- Example: Compute the minimum rank of the tree

T by

computing (T) = maximum multiplicity of eigenvalue 0:

1

2

3

4 5 6

8 7




21/69


References

pSimple GraphsSign PatternsMinimum Rank Problems for Tree Sign Patterns


T by


1

2

3

45 6

8 7




22/69


References

Simple GraphsSign PatternsMinimum Rank Problems for Tree Sign Patterns

Algorithm (for computing , and thus M and mr)While H = :


2. Set W = {w H:at most 1 component of TH w is not H-free}.3. Q = Q W .4. H = H W .

5. For each v H, ifdeg bTQv 2, remove v from H.




23/69


References



T by


1

2

3

4 56

8 7


Minimum Rank Problems for Simple GraphsMinimum Rank Problems for Simple TreesSi l G h


24/69


References




2. Set W = {w H:at most 1 component of TH w is not H-free}.3. Q = Q W .4. H = H W .



Minimum Rank Problems for Simple GraphsMinimum Rank Problems for Simple TreesSi l G h


25/69




T by


1

2

3

45 6

8 7


Minimum Rank Problems for Simple GraphsMinimum Rank Problems for Simple TreesSimple Graphs


26/69



Second Iteration:


1. Set

TH = the unique component of

T Q that contains an

H-vertex.

2. Set W = {w H:at most 1 component of TH w is not H-free}.

3. Q = Q W .4. H = H

W .





27/69




T by


1

2

3

45 6

7


C



28/69




T by


1

2

3

5 6

7


M i C l i P bl



29/69



- Example: Compute the minimum rank of the simple tree

T by

computing maximum multiplicity of eigenvalue 0:

- Q = {1, 2, 3, 5, 6, 7}, i.e., 6 vertices were deleted.- There are 18 paths.

- M(T) = 18 6 = 12.- mr(T) = 35 12 = 23.


M t i C l ti P bl



30/69


p pSign PatternsMinimum Rank Problems for Tree Sign Patterns

Trees are done, but work continues on

- minimum rank of arbitrary graphs-

- if graph has a cut-vertex, reduce the problem to the pieces[BFH04].

- new Colin de Verdiere type parameter used for computation ofminimum rank [BFH05],[HvdH06]

- matrices over fields other than R

- positive semidefinite minimum rank

- etc.


Matrix Completion Problems



31/69


p pSign PatternsMinimum Rank Problems for Tree Sign Patterns

Sign Patterns

- A sign pattern is an n n matrix Z = [zij] whose entries zijare elements of{+,, 0}.

- The qualitative class or sign pattern class of Z,

Q(Z) =

{A : sgn(aij) = zij, 1

i,j

n

}.

- The graph G(Z) of the symmetric n n sign pattern Z:- vertices 1, . . . , n- ij is an edge ofG(Z) if and only if zij = 0.

- The graph has loops.

- A symmetric sign pattern Z is a tree sign pattern (TSP) ifG(Z) is a tree.





32/69


Sign PatternsMinimum Rank Problems for Tree Sign Patterns

Minimum Rank Problems for Tree Sign Patterns

- For symmetric Z, S(Z) ={A : A is a symmetric matrix and for all i,j, sgn(aij) = zij}.

- The diagonal is restricted by the sign pattern.

- The minimum rank of Z is the minimum rank among matrices

having sign pattern Z.

- The symmetric minimum rank of symmetric Z is

mr(Z) = min

AS

(Z){rank(A)

}.

- For a symmetric tree sign pattern, minimum rank equalssymmetric minimum rank.



Minimum Rank Problems for Simple GraphsMinimum Rank Problems for Simple TreesSimple GraphsSi P


33/69



- The maximum multiplicity of an eigenvalue inS

(Z) is

M(Z) = maxAS(Z)

{mult(A)}.

- All positive (respectively, negative) eigenvalues have the same

maximum multiplicity.

- mr(Z) + M0(Z) = the size of Z.



Minimum Rank Problems for Simple GraphsMinimum Rank Problems for Simple TreesSimple GraphsSi P tt


34/69



Let Z be a (symmetric) sign pattern.- S(Z) (or G(Z)) allows eigenvalue if there is a matrix

A S(Z) having eigenvalue .- S(Z) allows eigenvalue 0 if and only if det Z is identically 0 or

has both positive and negative terms.



Minimum Rank Problems for Simple GraphsMinimum Rank Problems for Simple TreesSimple GraphsSign Patterns


35/69

pReferences


Algorithm (Computation ofM for TSP [DHHHW06] )

Let Z be a TSP and T = G(Z).Set Q = (set of vertices to be deleted).Set H = {vertices of T with degree 3} (candidates for deletion).While H = :


2. Set W = {w H:at most 1 component of TH w is not H-free}.

3. For each w W , if there are at least two H-free componentsof TH w that allow eigenvalue, then Q = Q {w}.4. H = H W .5. For each v H, ifdegTQv 2, remove v from H.

M

(Z)=(number of components of T

Q that allow) |

Q|.





36/69

ReferencesSign PatternsMinimum Rank Problems for Tree Sign Patterns

- Example: Compute the minimum rank of the sign pattern Z

by computing maximum multiplicity of eigenvalue 0:

1

2

3

56

4

8

9

7

10

11

12

1314

15

16

17 1819

20

_

_

_

_

_

+

+

_+

+

+

+

_





37/69




1

2

3

56

4

8

9

7

10

11

12

1314

15

16

17 18

1920

_

_

_

_

_

+

+

_+

+

+

+

_


Matrix Completion ProblemsR f



38/69


Algorithm (Computation ofM0)

While H = :1. Set TH = the unique component of T Q that contains an

H-vertex.

2. Set W =

{w

H:

at most 1 component of TH w is not H-free}.3. For each w W , if there are at least two H-free components

of TH w that allow eigenvalue 0, then Q = Q {w}.4. H = H

W .

5. For each v H, ifdegTQv 2, remove v from H.


Matrix Completion ProblemsR f



39/69

ReferencesS g atte sMinimum Rank Problems for Tree Sign Patterns



1

2

3

56

4

8

9

7

10

11

12

1314

15

16

17 18

1920

_

_

_

_

_

+

+

_+

+

+

+

_





40/69

Referencesg




H-vertex.

2. Set W =

{w

H:



W .






41/69

ReferencesMinimum Rank Problems for Tree Sign Patterns



1

2

3

56

4

8

9

7

10

11

12

1314

15

16

17 18

1920

_

_

_

_

_

+

+

_+

+

+

+

_





42/69




1

2

3

56

4

8

9

7

10

11

12

1314

15

16

17 18

1920

_

_

_

_

_

+

+

_+

+

+

+

_



Minimum Rank Problems for Simple GraphsMinimum Rank Problems for Simple TreesSimple GraphsSign PatternsMi i R k P bl f T Si P


43/69




1

2

3

56

4

8

9

7

10

11

12

1314

15

16

17 18

1920

_

_

_

_

_

+

+

_+

+

+

+

_



Minimum Rank Problems for Simple GraphsMinimum Rank Problems for Simple TreesSimple GraphsSign PatternsMi i R k P bl f T Si P tt


44/69




H-vertex.

2. Set W =

{w

H:



W .




Minimum Rank Problems for Simple GraphsMinimum Rank Problems for Simple TreesSimple GraphsSign PatternsMinimum Rank Problems for Tree Sign Patterns


45/69




1

2

3

5

8

9

7

10

11

12

1314

15

16

17 18

1920

_

_

_

_

_

+

+

+

+

+

+

_





46/69




H-vertex.

2. Set W =

{w

H:


of TH w) that allow eigenvalue 0, then Q = Q {w}.4. H = H W .5. For each v H, ifdegTQv 2, remove v from H.





47/69


- Example: Compute the minimum rank of the sign pattern Zby computing maximum multiplicity of eigenvalue 0:

1

2

3

5

8

9

7

10

11

12

1314

15

16

17 18

1920

_

_

_

_

_

+

+

+

+

+

+

_





48/69



1 3

5

8

9

7

10

11

12

1314

15

16

17 18

1920

_

_

_

_

_

+

+

+

+

+

+

_





49/69

g


- Q = {2, 4, 6}, i.e., 3 vertices were deleted.- There are 8 components that allow eigenvalue 0.

- M0(Z) = 8 3 = 5.- mr(Z) = 20 5 = 15.



DefinitionsGraph Theoretic TechniquesThe Positive (Semi)Definite Matrix Completion ProblemsThe (Strictly) Copositive Matrix Completion Problems


50/69


- A partial matrix is a square array in which some entries arespecified and others are not.

- A completion of a partial matrix is a choice of values for theunspecified entries.

Example:

B =

2 1 ? 01 2 2 ?

? 2 3 10 ? 1 1

A =

2 1 0 01 2 2 1

0 2 3 10 1 1 1

B is a partial matrix and A is a completion of B.





51/69

The following are equivalent for real matrices:

- A is symmetric and for all x Rn, x = 0, xTAx > 0(A is positive definite).

- A is symmetric and all eigenvalues are positive.

- A is symmetric and all principal minors are positive.

The following are equivalent for real matrices:

- A is symmetric and for all x Rn, xTAx 0(A is positive semidefinite).

- A is symmetric and all eigenvalues are nonnegative.

- A is symmetric and all principal minors are nonnegative.





52/69

Classes X of matrices to be discussed:

- positive definite matrices- positive semidefinite matrices

- strictly copositive matrices: A is strictly copositive if A issymmetric and for all x 0, x = 0, xTAx > 0

- copositive matrices: A is copositive if A is symmetric and forall x 0, xTAx 0

Example:

A = 2 1 3

1 1 23 2 2

is strictly copositive but not positive definite (any positive matrix isstrictly copositive).

Combinatorial Matrix TheoryMinimum Rank ProblemsMatrix Completion Problems

References



53/69

- A matrix completion problem asks whether a partial matrix, or

family of partial matrices with a given pattern of specifiedentries, has a completion of a specific type X, such as apositive definite matrix.

Example:

B =

2 1 ? 01 2 2 ?

? 2 3 10 ? 1 1

A =

2 1 0 01 2 2 1

0 2 3 10 1 1 1

Matrix A completes B to a positive semidefinite matrix.


References



54/69

- All classes X of matrices discussed are hereditary, i.e. if A isan X-matrix then every principal submatrix of A is anX-matrix.

- If X is hereditary, in order for a partial matrix B to have anX-completion, it is necessary that every fully specifiedprincipal submatrix of B is an X-matrix.

- A partial matrix B is a partial X-matrix if every fully specifiedprincipal submatrix of B is an X-matrix.

- a pattern of specified entries has the X-completion property if

every partial X-matrix B described by that pattern can becompleted to an X-matrix.


References



55/69

Graph Theoretic Techniques

- Graphs are used to describe the pattern of specified entriesclasses of for symmetric matrices such as positive(semi)definite and (strictly) copositive matrices; otherwisedigraphs are used.

- The specified entries in partial matrix B are represented by

edges in the graph G(B).Example:

B =

2 1 ? 01 2 2 ?

? 2 3 10 ? 1 1

1 2

34


References



56/69

Theorem ([GJSW84])

- A graph having a loop at every vertex has the positive definitecompletion property if and only if it is chordal (any cycle oflength 4 has a chord).

- A graph has the positive definite completion property if and

only if the subgraph induced by the vertices with loops hasthe positive definite completion property.

Example:

has the positive definite completion property.


References



57/69

Theorem ([GJSW84])A graph has the positive semidefinite completion property if andonly if each component has all loops or none, and componentswith loops are chordal.

Example:

does not have the positive semidefinite completion property.


References



58/69

The (Strictly) Copositive Matrix Completion Problems

- A is strictly copositive if A is symmetric and for allx 0, x = 0, xTAx > 0.

- A is copositive if A is symmetric and for all x 0, xTAx 0.Example:

A =

1 1 1 1

1 1 1 11 1 1 1

1 1 1 1

is copositive but not strictly copositive, and not positivesemidefinite.


References



59/69

- The partial matrix B is a partial strictly copositive matrix if

every fully specified principal submatrix of B is a strictlycopositive matrix.

- The partial matrix B is a partial copositive matrix if everyfully specified principal submatrix of B is a copositive matrix.

Example:

B =

3 1 x13 11 1 2 x24

x13 2 1 11 x24 1 2

is a partial strictly copositive matrix.


References



60/69

Theorem ([HJR05])

Let B be a partial copositive matrix with every diagonal entryspecified. For each pair of unspecified off-diagonal entries, setxij = xji =

biibjj. The resulting matrix is copositive, and is

strictly copositive if B is a partial strictly copositive matrix.

Example:

B =

3 1 x13 11 1 2 x24

x13

2 1

11 x24 1 2

A =

3 1

3 1

1 1 2

23 2 1

1

1 2 1 2

A completes B to a strictly copositive matrix.


References



61/69

Theorem ([H06])

Let B = x11 bTb B1

be a partial strictly copositive n n matrixhaving all entries except the 1,1-entry specified. Let be avector norm. Complete B to a strictly copositive matrix bychoosing a value for x

11as follows:

1. = miny0,y=1 bTy.

2. = miny0,||y||=1 yTB1y.

3. x11 > 2

.


References



62/69

Corollary

Every partial strictly copositive matrix can be completed to astrictly copositive matrix.


References



63/69

Example: The partial matrix

B =

x11 5 1 x14 x15 x165 1 2 x24 x25 1

1 2 5 1 1 1x14 x24 1 1 x45 1x15 x25

1 x45 x55 x56

x16 1 1 1 x56 3

is a partial strictly copositive matrix.

Select index 5. The only principal submatrix completed by a choice

of b55 is B[{3, 5}].Any value that makes 5x55 > (1)2 will work.Choose x55 = 1.


References



64/69

Example: The partial matrix

B =

x11 5 1 x14 x15 x165 1 2 x24 x25 1

1 2 5 1 1 1x14 x24 1 1 x45 1x15 x25

1 x45 x55 x56

x16 1 1 1 x56 3

is a partial strictly copositive matrix.Select index 1. The only principal submatrices completed by achoice of b11 are principal submatrices of

B[{1, 2, 3}] = x11 5 15 1 2

1 2 5

.


References



65/69

B[{1, 2, 3}] = x11

5 1

5 1 21 2 5

Using 1:

1. = min||y||1=1 bTy = 5.

2. = min||y||1=1 yTB[{2, 3}]y = 110 .

3. Choose x11 >2

; choose b11 = 256.


References


6


66/69

b11 = 256, b55 = 1.

256 5 1 x14 x15 x16

5 1

2 x24 x25 1

1 2 5 1 1 1x14 x24 1 1 x45 1x15 x25 1 x45 1 x56x16 1 1 1 x56 3

Set xij = xji = biibjj.

256 5 1 16 16 16 35 1 2 1 1 1

1 2 5 1 1 1

16 1 1 1 1 116 1 1 1 1 316

3 1 1 1 3 3

is a strictly copositive matrix.


References



67/69

It is not true that every partial copositive matrix can be completedto a copositive matrix.

Example: B =

x11 11 0

is a partial copositive matrix that

cannot be completed to a copositive matrix.

Choose a value for x11 (> 0).If x11 > 0, then for the vector x = [1, x11]

T, xTBx = x11.


References


68/69

[BFH04] F. Barioli, S. Fallat, and L. Hogben. Computation ofminimal rank and path cover number for graphs. Linear

Algebra and Its Applications, 392: 289303, 2004.

[BFH05] F. Barioli, S. Fallat, and L. Hogben. A variant on the graphparameters of Colin de Verdiere: Implications to theminimum rank of graphs. Electronic Journal of Linear

Algebra, 13: 387404, 2005.

[DHHHW06] L.M. DeAlba, T.L. Hardy, I.R. Hentzel, L. Hogben, A.Wangsness. Minimum Rank and Maximum EigenvalueMultiplicity of Symmetric Tree Sign Patterns. To appear inLinear Algebra and Its Applications.

[GJSW84] R. Grone, C.R. Johnson, E.M. Sa, and H. Wolkowicz.Positive definite completions of partial Hermitian matrices.Linear Algebra and Its Applications, 58:109124, 1984.




69/69

[H06] L. Hogben. Completions of partial strictly copositivematrices omitting some diagonal entries. To appear in

Linear Algebra and Its Applications.

[HvdH06] L. Hogben and H. van der Holst. Forbidden minors for theclass of graphs G with (G) 2. To appear in LinearAlgebra and Its Applications. PDF provided.

[HJR05] L. Hogben, C. R. Johnson and R. Reams. The copositivematrix completion problem. Linear Algebra and ItsApplications, 408:207-211, 2005.

[JLD99] C. R. Johnson and A. Leal Duarte. The maximum

multiplicity of an eigenvalue in a matrix whose graph is atree. Linear and Multilinear Algebra 46: 139144, 1999.

introduction to combinatorial matrix theory

Documents