Matrix Completion Problems for Various Classes of P-Matrices Leslie Hogben Department of Mathematics, Iowa State University, Ames, IA 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(  ). A principal minor is the determinant of a principal submatrix. The matrix M  R n  n is a P-matrix (P 0 - 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

nonnegative, which requires a ij > 0 for all i,j positive, which requires a ij > 0 for all i,j weakly sign symmetric, which requires a ij a ji > 0 for each pair i,j sign symmetric, which requires a ij a ji > 0 or a ij = 0 = a ji for each pair symmetric, which requires a ij = a ji for all i,j (symmetric P- matrices are the same as positive definite matrices) C is a weakly sign symmetric P-matrix, but is not sign symmetric. D is a positive P-matrix. Additional subclasses of P-matrices (P 0 -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.

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.

The partial matrix A is a partial P 0 -matrix if every fully specified principal submatrix is a P 0 -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 P 0 -matrix if every fully specified principal submatrix is a nonnegative P 0 -matrix and every specified entry is nonnegative. The partial matrix A is a partial (weakly) sign symmetric P- matrix (P 0 -matrix) if every fully specified principal submatrix is a (weakly) sign symmetric P- matrix (P 0 -matrix).

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 (P 0 -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.

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 a ij of A is specified exactly when (i,j)  E. The partial symmetric matrix A specifies the graph G if the entry a ij 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 Terminology The order of a (di)graph is the number of vertices. A sub(di)graph of the (di)graph G = (V G,E G ) is a (di)graph H = (V H,E H ), where V H  V G and E H  E G. If W  V G, the sub(di)graph of G induced by W,  W  is the digraph (W,E W ) with E W = E G  (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 v 1, v 2,..., v k,v k+1 in V such that for i=1,…,k the arc (v i,v i+1 )  E and all vertices are distinct except possibly v 1 = v k+1. A semipath in a digraph G = (V,E) is sequence of vertices v 1, v 2,..., v k,v k+1 in V such that for i=1,…,k the arc (v i,v i+1 )  E or (v i+1,v i )  E and all vertices are distinct except possibly v 1 = v k+1. a 3-cycle a path of length 3 If v 1 = v k+1 the (semi)path is a (semi)cycle. The length of the (semi)path v 1, v 2,..., v k,v k+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.

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.

For most of the classes, the same idea works as long as each strongly connected induced subdigraph has X-completion. 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, P 0 -matrices, nonnegative P 0 -matrices, weakly sign symmetric P 0 -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: , 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 P 0 -matrices nonnegative P 0 -matrices weakly sign symmetric P 0 -matrices sign symmetric P 0 -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 P 0 -matrices nonnegative P 0 -matrices weakly sign symmetric P 0 -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 X 0 are referred to as a pair of  /  0 -classes if 1. Any partial X-matrix is a partial X 0 -matrix. 2. For any X 0 -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 P 0 -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

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.

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 P 0 -completion has P-completion. Any digraph that has weakly sign symmetric P 0 -completion has weakly sign symmetric P-completion. Any digraph that has sign symmetric P 0 -completion has sign symmetric P-completion. Any digraph that has nonnegative P 0 -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.

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) P 0 - matrix, the graph of whose specified entries is a double triangle such that A cannot be completed to a nonnegative ((weakly) sign symmetric) P 0 -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. 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 -3

Definition The graph G is a minimally chordal Hamiltonian graph if 1.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. 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.

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 P o - Matrix Completion Problem, Electronic Journal of Linear Algebra, 9:1-20, 2003] Let G be a symmetric n-cycle. Every partial P 0 -matrix specifying G can be completed to a P 0 -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 P 0 -matrix completion problem. Electronic Journal of Linear Algebra, 10:46–59, 2003] Let G be a symmetric n-cycle. Every partial nonnegative P 0 -matrix specifying G can be completed to a nonnegative P 0 -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 P 0 -matrix (weakly sign symmetric P- matrix, sign symmetric P-matrix) specifying G can be completed to a weakly sign symmetric P 0 -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 P 0 -matrix Q specifying G cannot be completed to a sign symmetric P 0 -matrix.

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 P 0 -matrix specifying G can be completed to a sign symmetric P 0 -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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