Matrix Completion Problems forVarious Classes of P-Matrices

Leslie Hogben

Department of Mathematics,

Iowa State University, Ames, IA 50011

LHogben@iastate.edu

http://www.math.iastate.edu/lhogben/MC/

Colloquium

Department of Mathematics

The College of William and Mary

October 31, 2003

All matrices discussed are real and square.

The principal submatrix of the n n matrix A defined by the subset of {1,...,n} is the submatrix of A that lies in the rows and columns indexed by it is denoted by A().

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A=

2 −1 0

3 1 −4

5 −1 5

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A principal minor is the determinant of a principal submatrix. The matrix M Rnn is a P-matrix (P0-matrix) if every principal minor of M is positive (nonnegative).

A is a P-matrix, but B is not since Det B(1,2) = -1

€

B=

2 −1 0

−3 1 −4

5 −1 5

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⎦

⎥ ⎥ ⎥

€

A(1,3) =2 0

5 5

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⎦ ⎥ ⎥

• nonnegative, which requires aij > 0 for all i,j • positive, which requires aij > 0 for all i,j• weakly sign symmetric, which requires aij aji > 0 for each

pair i,j• sign symmetric, which requires aij aji > 0 or aij = 0 = aji for

each pair• symmetric, which requires aij = aji for all i,j (symmetric P-

matrices are the same as positive definite matrices)

€

C =

2 −1 0

−3 2 −4

5 −1 5

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⎥ ⎥ ⎥

C is a weakly sign symmetric P-matrix, but is not sign symmetric. D is a positive P-matrix.

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D=

2 1 2

1 3 1

3 2 5

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⎥ ⎥ ⎥

Additional subclasses of P-matrices (P0-matrices) are obtained by imposing various restrictions on the signs of entries.

A partial matrix is a matrix in which some entries are specified and others are not. E is a partial matrix.

A completion of a partial matrix is a choice of values for the unspecified entries.

The matrix A is a completion of E. But of course we want special properties in the completion. In fact, A completes E to a P-matrix.

But there is no hope of completing F to a P-matrix.

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E =

2 −1 0

3 1 ?

? −1 5

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⎥ ⎥ ⎥

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A=

2 −1 0

3 1 −4

5 −1 5

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⎤

⎦

⎥ ⎥ ⎥

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F =

2 1 0

3 1 ?

? 5 5

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⎥ ⎥ ⎥

Most generally, a matrix completion problem asks whether the unspecified entries can be chosen so the completed matrix is of a desired type (e.g., P-matrix) and/or asks for a “best” completion of this type.

A combinatorial matrix completion problem asks whether every partial X-matrix with the given pattern of unspecified entries can be completed to an X -matrix (for a given class X of matrices).

The partial matrix M is a partial P-matrix if every fully specified principal submatrix is a P- matrix.

E is a partial P-matrix, but F is not.

€

F =

2 1 0

3 1 ?

? 5 5

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⎥ ⎥ ⎥

€

E =

2 −1 0

3 1 ?

? −1 5

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⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

The partial matrix A is a partial P0-matrix if every fully specified principal submatrix is a P0-matrix.

The partial matrix A is a partial positive (nonnegative) P-matrix if every fully specified principal submatrix is a positive (nonnegative) P-matrix and every specified entry is positive (nonnegative).

The partial matrix A is a partial nonnegative P0-matrix if every fully specified principal submatrix is a nonnegative P0-matrix and every specified entry is nonnegative.

The partial matrix A is a partial (weakly) sign symmetric P-matrix (P0-matrix) if every fully specified principal submatrix is a (weakly) sign symmetric P- matrix (P0-matrix).

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L =

2 1 0

1 1 ?

? 3 5

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⎥ ⎥ ⎥

E is a partial P-matrix but is not partial weakly sign symmetric.

L is a partial nonnegative P-matrix (also partial sign symmetric, but not partial positive).

We are looking for patterns of specified positions that have the property that any partial (symmetric, nonnegative, positive, (weakly) sign symmetric) P-matrix (P0-matrix) can be completed to the same type.

A pattern for P-matrices has this property if and only if the principal subpattern determined by the specified diagonal positions does. So we assume all diagonal positions are specified.

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E =

2 −1 0

3 1 ?

? −1 5

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⎥ ⎥ ⎥

The pattern of specified positions in a partial matrix (that has all diagonal entries specified) can be described by a graph (for a symmetric matrix) or more generally by a digraph.

A graph G = (V,E) is a finite set of positive integers V, whose members are called vertices, and a set E of pairs {v,u} of distinct vertices, called edges.

a graph a digraph

A directed graph (digraph) G = (V,E) is a finite set of positive integers V, and a set E of ordered pairs (v,u) of distinct vertices, called arcs.

All partial matrices discussed have all diagonal entries specified. The partial matrix A specifies the digraph G if the entry aij of A is specified exactly when (i,j) E.

€

K =

1 −1 0 ?

−2 5 2 0

3 1 2 4

−1 ? ? 1

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The partial symmetric matrix A specifies the graph G if the entry aij of A is specified exactly when {i,j} E.

The (di)graph G is said to have X-completion if every partial X-matrix specifying G can be completed to an X-matrix.

We wish to determine the (di)graphs that have X-completion.

Graph TerminologyThe order of a (di)graph is the number of vertices. A sub(di)graph of the (di)graph G = (VG,EG) is a (di)graph H = (VH,EH), where VH VG and EH EG. If W VG, the sub(di)graph of G induced by W, W is the digraph (W,EW) with EW = EG (W W). A sub(di)graph induced by a subset of vertices is also called an induced sub(di)graph.

H is an induced subdigraph of G

A path in a digraph G = (V,E) is sequence of vertices v1, v2, ..., vk,vk+1 in V such that for i=1,…,k the arc (vi,vi+1) E and all vertices are distinct except possibly v1 = vk+1.

A semipath in a digraph G = (V,E) is sequence of vertices v1, v2, ..., vk,vk+1 in V such that for i=1,…,k the arc (vi,vi+1) E or (vi+1,vi) E and all vertices are distinct except possibly v1 = vk+1.

a 3-cycle a path of length 3

If v1 = vk+1 the (semi)path is a (semi)cycle.The length of the (semi)path v1, v2, ..., vk,vk+1 is k.

A graph (digraph) is connected if there is a path (semipath) from any vertex to any other vertex (a (di)graph of order 1 is connected); otherwise it is disconnected. A component of a (di)graph is a maximal connected sub(di)graph.

connected but not strongly connected strongly connected

A digraph is strongly connected if there is a path from any vertex to any other vertex.

€

ˆ M =

a11 a12 0 0

a21 a22 0 0

0 0 a33 0

0 0 a43 a44

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H1

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H1

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H2

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H2

Suppose the partial P-matrix M has digraph G and G has two components, and , each of which has P-completion. Then M can be completed to a P-matrix by completing the diagonal blocks corresponding to and and then setting the remaining entries to 0.

This technique works for all the subclasses we discuss here, so the problem of determining which (di)graphs have completion is reduced to determining which connected (di)graphs have completion.

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M =

a11 a12 ? ?

a21 a22 ? ?

? ? a33 ?

? ? a43 a44

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For most of the classes, the same idea works as long as each strongly connected induced subdigraph has X-completion.

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M =

a11 a12 a12

? a22 a23

? a32 a33

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⎥ ⎥ ⎥

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ˆ M =

a11 a12 a12

0 a22 a23

0 a32 a33

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⎥ ⎥ ⎥

In general, if a (di)graph has X-completion, so does any induced subdigraph.

For X any of the classes of P-matrices, positive P-matrices, nonnegative P-matrices, (weakly) sign symmetric P-matrices, P0-matrices, nonnegative P0-matrices, weakly sign symmetric P0-matrices a digraph has X-completion if and only if every strongly connected induced subdigraph has X-completion.

In a graph, a chord of a cycle is an edge joining two non-consecutive vertices of the cycle. A graph G is chordal if any cycle of length > 3 in G has a chord.

a chordal graph not chordal

A cycle of a graph G is a Hamiltonian cycle of G if it contains all the vertices of G. The cycle 1,2,3,4,1 is a Hamilton cycle in these graphs.

More graph terminology

A digraph is nonseparable if it is connected and has no cut-vertices. A block of a digraph is a subdigraph that is nonseparable and is maximal with respect to this property.

A (sub)digraph is called a clique if it contains all possible arcs between its vertices. A digraph is 1-chordal if every block is a clique.

A cut-vertex is a vertex whose deletion disconnects the component containing the vertex.

2 blocks, 1 cut vertex

Theorem [R. Grone, C. R. Johnson, E. M. Sá, and H. Wolkowicz, Positive Definite Completions of Partial Hermitian Matrices, Linear Algebra and Its

Applications 58:109-124, 1984] A graph has positive definite completion if and only if it is chordal.

(Recall positive definite matrix = symmetric P-matrix)

This was the first result classifying patterns (or equivalently graphs) as to whether every partial X-matrix specifying the graph could be completed to an X-matrix.

Since then progress has been made on the matrix completion problem for several individual classes. For example:

In the last few years there have been two additional developments:

•multi class theorems

•relationship theorems

Theorem [C. R. Johnson and B. K. Kroschel, The Combinatorially Symmetric P-Matrix Completion Problem, Electronic Journal of Linear Algebra

1: 59-63, 1996] Every graph has P-completion. That is, any partial P-matrix whose specified positions are symmetrically placed can be completed to a P-matrix.

Theorem [S. M. Fallat, C. R. Johnson, J. R. Torregrosa, and A. M. Urbano. P-matrix completions under weak symmetry assumptions. Linear Algebra and Its Applications

312:73–91, 2000]. Let X be any of the classes • P-matrices• nonnegative P-matrices• positive P-matrices• weakly sign symmetric P-matrices• sign symmetric P-matrices • P0-matrices• nonnegative P0-matrices• weakly sign symmetric P0-matrices• sign symmetric P0-matrices

Let G be a 1-chordal graph. Then G has X-completion, i.e., any partial X-matrix specifying G can be completed to an X-matrix.

Multi-Class Theorems

Theorem [L. Hogben, Graph theoretic methods for matrix completion

problems. Linear Algebra and ItsApplications 328:161–202, 2001] Let X be any of the classes

• P-matrices• nonnegative P-matrices• positive P-matrices• weakly sign symmetric P-matrices• sign symmetric P-matrices • P0-matrices• nonnegative P0-matrices• weakly sign symmetric P0-matrices

A digraph G has X-completion if and only if every nonseparable strongly connected induced subdigraph has X-completion.

The classes of matrices X and X0 are referred to as a pair of /0-classes if

1. Any partial X-matrix is a partial X0-matrix.2. For any X0-matrix A and > 0, A + I is a X-matrix.3. For any partial X-matrix A, there exists > 0 such that A - is a partial X-matrix (where is the partial identity matrix specifying the same digraph as A).

The classes P-matrices and P0-matrices are a pair of /0-classes.

Theorem [L. Hogben. Matrix Completion Problems for Pairs of Related Classes

of Matrices, to appear in Linear Algebra and Its Applications]. For a pair of /0-classes, if a digraph has 0-completion then it must also have -completion.

Relationship Theorems

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Theorem For a pair of /0-classes, if a digraph has 0-completion then it must also have -completion.

Proof Let Q be a pattern that has 0-completion, and let A be a partial -matrix specifying Q. Let be the partial identity matrix specifying the pattern Q. There is a > 0 such that B = A - is a partial -matrix, and hence a partial 0-matrix. Since Q has 0-completion, B can be completed to a 0-matrix . Then = + I is a -matrix that completes A. Thus Q has -completion.

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˜ I

€

˜ I

€

ˆ A

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ˆ B

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ˆ B

Corollary [L. Hogben. Matrix Completion Problems for Pairs of Related

Classes of Matrices, to appear in Linear Algebra and Its Applications]. •Any digraph that has P0-completion has P-completion. •Any digraph that has weakly sign symmetric P0-completion has weakly sign symmetric P-completion. •Any digraph that has sign symmetric P0-completion has sign symmetric P-completion. •Any digraph that has nonnegative P0-completion has nonnegative P-completion.

Theorem Let G be a digraph that has nonnegative P-completion. Then G has positive P-completion.

Proof. Let A be a partial positive P-matrix specifying G. The matrix A is a partial nonnegative P-matrix specifying G, and so can be completed to nonnegative P-matrix . The only reason might not be a positive P-matrix is if some entries (that were originally unspecified) are zero. Since there are only finitely many principal minors of and these are continuous functions of the entries of , we can slightly perturb zero entries while maintaining all principal minors positive. Thus can be converted into a positive P-matrix that completes A.

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ˆ A

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ˆ A

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ˆ A

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ˆ A

a double triangle

Digraphs that prevent completion

Theorem [S. M. Fallat, C. R. Johnson, J. R. Torregrosa, and A. M. Urbano. 2000]. There exists a partial positive (nonnegative, (weakly) sign symmetric) P-matrix, the graph of whose specified entries is a double triangle such that A cannot be completed to a positive (nonnegative, (weakly) sign symmetric) P-matrix. There exists a partial nonnegative ((weakly) sign symmetric) P0-matrix, the graph of whose specified entries is a double triangle such that A cannot be completed to a nonnegative ((weakly) sign symmetric) P0-matrix.

Example [C. R. Johnson and B. K. Kroschel, The Combinatorially Symmetric P-Matrix Completion Problem, Electronic Journal of Linear Algebra 1: 59-63, 1996]

The partial P-matrix A (whose graph is shown) cannot be completed to a P-matrix.

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A=

1 −1 1 1

2 1 −1 1

0 1 1 2

y −10 −1 1

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In fact, the 2,3 and 3,2 entries are irrelevant. It is the two partial 3 x 3 matrices that give conflicting requirements:

A(1,2,4) requires y < -7/2 and A(1,3,4) requires y > -3

Definition The graph G is a minimally chordal Hamiltonian graph if1. G has a Hamiltonian cycle H.2. G is chordal.3. If any non-empty set S of chords of H is removed from G, the

resulting graph is not chordal.

This example generalizes to a family of digraphs that prevent completion:

Definition A digraph G is a minimally chordal symmetric-Hamiltonian digraph if 1. The underlying graph G’ of G is a minimally chordal

Hamiltonian graph.2. Each arc corresponding to part of the unique

Hamiltonian cycle of G’ is symmetric in G.3. Each arc corresponding to a chord of the unique

Hamiltonian cycle of G’ is asymmetric in G.

Theorem [L. Hogben, J. Evers, S. M. Shaner, The Positive and Nonnegative P-

matrix Completion Problems, preprint] For any pattern that contains all diagonal positions and whose digraph is a minimally chordal symmetric-Hamiltonian digraph with at least 4 vertices, there exists a partial positive P-matrix that cannot be completed to a P-matrix.

Use the matrices etc.

with a, b, u, v, q, close to 0 to force x close to 0,

i.e., the entry 1-x close to 1.

€

1 1−u 1−q

1−a 1 1−v

1−x 1−b 1

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Work in from the highest triangle label toward the root (labeled zero in the diagram).

Use the matrix

at the root, which forces y < 0.464, to get a contradictory result.

€

1 0.1 1

0.4 1 0.9

0.8 y 1

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Theorem [S. M. Fallat, C. R. Johnson, J. R. Torregrosa, and A. M. Urbano. P-matrix completions under weak symmetry assumptions. Linear Algebra and Its

Applications 312:73–91, 2000]. If A is a partial positive (nonnegative) P-matrix, the digraph of whose specified entries is a symmetric n-cycle, then A can be completed to a positive (nonnegative) P-matrix.

a symmetric 6-cycle

Theorem [J. Y. Choi, L. M. DeAlba,, L. Hogben, M. S. Maxwell, and A. Wangsness,

The Po - Matrix Completion Problem, Electronic Journal of Linear Algebra, 9:1-20, 2003] Let G be a symmetric n-cycle. Every partial P0-matrix specifying G can be completed to a P0-matrix if and only if n 4.

Symmetric Cycles

Theorem [J. Y. Choi, L. M. DeAlba, L. Hogben, B. M. Kivunge, S. K. Nordsrom, and M. Shedenhelm. The nonnegative P0-matrix completion problem. Electronic

Journal of Linear Algebra, 10:46–59, 2003] Let G be a symmetric n-cycle. Every partial nonnegative P0-matrix specifying G can be completed to a nonnegative P0-matrix if and only if n 4.

Theorem [L. DeAlba, T. L.Hardy, L. Hogben, and A. Wangsness, The (weakly) sign symmetric P-matrix completion problems, submitted to Electronic

Journal of Linear Algebra] Let G be a symmetric n-cycle. Every partial weakly sign symmetric P0-matrix (weakly sign symmetric P-matrix, sign symmetric P-matrix) specifying G can be completed to a weakly sign symmetric P0-matrix (weakly sign symmetric P-matrix, sign symmetric P-matrix) if and only if n 4, 5.

Theorem [S. M. Fallat, C. R. Johnson, J. R. Torregrosa, and A. M. Urbano. P-matrix completions under weak symmetry assumptions. Linear Algebra and Its Applications

312:73–91, 2000]. Let G be a symmetric n-cycle (n > 4). The partial sign symmetric P0-matrix Q specifying G cannot be completed to a sign symmetric P0-matrix.

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Q=

1 1 ? L ? −1

1 1 1 L ? ?

? 1 1 L ? ?

M M M O M M

? ? ? L 1 1

−1 ? ? L 1 1

⎡

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⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

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⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Theorem [S. M. Fallat, C. R. Johnson, J. R. Torregrosa, and A. M. Urbano,

2000], [L. Hogben, Graph theoretic methods, 2001] . The digraph G has the property that any partial sign symmetric P0-matrix specifying G can be completed to a sign symmetric P0-matrix if and only if G is symmetric and 1-chordal.

Recall [FJTU] showed 1-chordal is ok, and the symmetric cycle and double triangle do not work.

The above example shows a symmetric pattern is required.

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A=0 1

? 0

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http://www.math.iastate.edu/lhogben/research/

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