Partial matrices whose completions have ranks bounded below

A partial matrix over a field F is a matrix whose entries are either elements of F or independent indeterminates. A completion of such a partial matrix is obtained by specifying values from F for the indeterminates. We determine the maximum possible number of indeterminates in a partial m×n matrix whose completions all have rank at least equal to a particular k, and we fully describe those examples in which this maximum is attained. Our main theoretical tool, which is developed in Section 2, is a duality relationship between affine spaces of matrices in which ranks are bounded below and affine spaces of matrices in which the (left or right) nullspaces of elements possess a certain covering property.     

ACI-matrices all of whose completions have the same rank

We characterize the ACI-matrices all of whose completions have the same rank, determine the largest number of indeterminates in such partial matrices of a given size, and determine the partial matrices that attain this largest number.     

On completions of symmetric and antisymmetric block diagonal partial matrices

A partial matrix is a matrix where only some of the entries are given. We determine the maximum rank of the symmetric completions of a symmetric partial matrix where only the diagonal blocks are given and the minimum rank and the maximum rank of the antisymmetric completions of an antisymmetric partial matrix where only the diagonal blocks are given.     

Partial matrices all of whose completions have the same spectrum

We characterize the square partial matrices over a field all of whose completions have the same spectrum, and determine the maximum number of indeterminates in such partial matrices of a given order as well as the matrices that attain this maximum number.     

Positive definite completions of partial Hermitian matrices

The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of the specified entries is chordal, a positive definite completion necessarily exists. Furthermore, if this graph is not chordal, then examples exist without positive definite completions. In case a positive definite completion exists, there is a unique matrix, in the class of all positive definite completions, whose determinant is maximal, and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial Hermitian matrix. Additional observations regarding positive definite completions are made.     

An extremal sparsity property of the Jordan canonical form

We prove that among all the matrices that are similar to a given square complex matrix, the Jordan canonical form has the largest number of off-diagonal zero entries. We also characterize those matrices that attain this largest number.     

The positive definite completion problem revisited

In [R. Grone, C.R. Johnson, E. Sa, H. Wolkowicz, Positive definite completions of partial Hermitian matrices, Linear Algebra Appl. 58 (1984) 109-124] the positive definite (semi-) completion problem in which the underlying graph is chordal was solved. For the positive definite case, the process was constructive and the completion was obtained by completing the partial matrix an entry at a time. For the positive semidefinite case, they obtained completions of a particular sequence of partial positive definite matrices with the same underlying graph and noted that there is a convergent subsequence of these completions that converges to the desired completion. Here, in the chordal case, we provide a constructive solution, based entirely on matrix/graph theoretic methods, to the positive (semi-)definite completion problem. Our solution associates a specific tree (called the “clique tree” [C.R. Johnson, M. Lundquist, Matrices with chordal inverse zero-patterns, Linear and Multilinear Algebra 36 (1993) 1-17]) with the (chordal) graph of the given partial positive (semi-)definite matrix. This tree structure allows us to complete the matrix a “block at a time” as opposed to an “entry at a time” (as in Grone et al. (1984) for the positive definite case). In Grone et al. (1984), using complex analytic techniques, the completion for the positive definite case was shown to be the unique determinant maximizing completion and was shown to be the unique completion that has zeros in its inverse in the positions corresponding to the unspecified entries of the partial matrix. Here, we show the same using only matrix/graph theoretic tools.     

Superoptimal completions of triangular matrices

We construct the unique completion of a partial triangular matrix with compact operator entries that has the property that its sequence of singular values is minimal in lexicographical order among all completions. In addition some partial results regarding the singular values of this superoptimal completion are presented.The research was done while the author visited the Department of Mathematics at the George Washington University.Supported by the College of William and Mary     

Matrix Completions and Chordal Graphs

In a matrix-completion problem the aim is to specifiy the missing entries of a matrix in order to produce a matrix with particular properties. In this paper we survey results concerning matrix-completion problems where we look for completions of various types for partial matrices supported on a given pattern. We see that the existence of completions of the required type often depends on the chordal properties of graphs associated with the pattern.     

Subdirect sums of P(P0)-matrices and totally nonnegative matrices

In this paper, the problem of when the sub-direct sum of two strictly diagonally dominant P-matrices is a strictly diagonally dominant P-matrix is studied. In particular, it is shown that the subdirect sum of overlapping principal submatrices of strictly diagonally dominant P-matrices is a strictly diagonally dominant P-matrix. It is also established that the 2-subdirect sum of two totally nonnegative matrices is a totally nonnegative matrix under some conditions. It is obtained that a partial totally nonnegative matrix, whose graph of the specified entries is a monotonically labeled 2-chordal graph, has a totally nonnegative completion. Finally, a positive answer to the question (IV) in Fallat and Johnson [Shaun M. Fallat, C.R. Johnson, J.R. Torregrosa, A.M. Urbano, P-matrix completions under weak symmetry assumptions, Linear Algebra Appl. 312 (2000) 73-91] is given for P₀-matrices.     

Intrinsic products and factorizations of matrices

We say that the product of a row vector and a column vector is intrinsic if there is at most one nonzero product of corresponding coordinates. Analogously we speak about intrinsic product of two or more matrices, as well as about intrinsic factorizations of matrices. Since all entries of the intrinsic product are products of entries of the multiplied matrices, there is no addition. We present several examples, together with important applications. These applications include companion matrices and sign-nonsingular matrices.     

Completion problem with partial correlation vines

This paper extends the results in [D. Kurowicka, R.M. Cooke, A parametrization of positive definite matrices in terms of partial correlation vines, Linear Algebra Appl. 372 (2003) 225-251]. We show that a partial correlation vine represents a factorization of the determinant of the correlation matrix. We show that the graph of an incompletely specified correlation matrix is chordal if and only if it can be represented as an m-saturated incomplete vine, that is, an incomplete vine for which all edges corresponding to membership-descendents (m-descendents for short) of a specified edge are specified. This enables us to find the set of completions, and also the completion with maximal determinant for matrices corresponding to chordal graphs.     

Multidimensional Permanents and an Upper Bound on the Number of Transversals in Latin Squares

The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. A nonnegative matrix whose every 1‐dimensional plane sums to 1 is called polystochastic. A latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and each column. A transversal of such a square is a set of n entries such that no two entries share the same row, column, or symbol. Let T(n) be the maximum number of transversals over all latin squares of order n. Here, we prove that over the set of multidimensional polystochastic matrices of order n the permanent has a local extremum at the uniform matrix for whose every entry is equal to . Also, we obtain an asymptotic value of the maximal permanent for a certain set of nonnegative multidimensional matrices. In particular, we get that the maximal permanent of polystochastic matrices is asymptotically equal to the permanent of the uniform matrix, whence as a corollary we have an upper bound on the number of transversals in latin squares      

The copositive completion problem: Unspecified diagonal entries

In [L. Hogben, C.R. Johnson, R. Reams, The copositive matrix completion problem, Linear Algebra Appl. 408 (2005) 207-211] it was shown that any partial (strictly) copositive matrix all of whose diagonal entries are specified can be completed to a (strictly) copositive matrix. In this note we show that every partial strictly copositive matrix (possibly with unspecified diagonal entries) can be completed to a strictly copositive matrix, but there is an example of a partial copositive matrix with an unspecified diagonal entry that cannot be completed to a copositive matrix.     

Characterization of full rank ACI-matrices over fields

An ACI-matrix over a field F$\mathbb{F}$ is a matrix whose entries are polynomials with coefficients on F$\mathbb{F}$, the degree of these polynomials is at most one in a number of indeterminates, and where no indeterminate appears in two different columns. In 2011 Huang and Zhan characterized the m×n$m\times n$ ACI-matrices such that all its completions have rank equal to min{m,n}$\min \{m,n\}$ whenever |F|?max{m,n+1}$|\mathbb{F}|?\max \{m,n+1\}$. We will give a characterization for arbitrary fields by introducing two classes of ACI-matrices: the maximal and the minimal full rank ACI-matrices.     

Hermitian positive semidefinite matrices whose entries are 0 or 1 in Modulus

We show that a matrix is a Hermitian positive semidefinite matrix whose nonzero entries have modulus 1 if and only if it is similar to a direct sum of all I's matrices and a 0 matrix via a unitary monomial similarity. In particular, the only such nonsingular matrix is the identity matrix and the only such irreducible matrix is similar to an all l's matrix by means of a unitary diagonal similarity. Our results extend earlier results of Jain and Snyder for the case in which the nonzero entries (actually) equal 1. Our methods of proof, which reiy on the so called principal submatrix rank property, differ from the approach used by Jain and Snyder.     

Required nonzero patterns for nonsingular sign regular matrices

An n×n real matrix is called sign regular if, for each k(1?k?n), all its minors of order k have the same nonstrict sign. The zero entries which can appear in a nonsingular sign regular matrix depend on its signature because the signature can imply that certain entries are necessarily nonzero. The patterns for the required nonzero entries of nonsingular sign regular matrices are analyzed.     

Theorems on partitioned matrices revisited and their applications to graph spectra

Some old results about spectra of partitioned matrices due to Goddard and Schneider or Haynsworth are re-proved. A new result is given for the spectrum of a block-stochastic matrix with the property that each off-diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off-diagonal entries. The result is applied to the study of the spectra of the usual graph matrices by partitioning the vertex set of the graph according to the neighborhood equivalence relation. The concept of a reduced graph matrix is introduced. The question of when n-2 is the second largest signless Laplacian eigenvalue of a connected graph of order n is treated. A recent conjecture posed by Tam, Fan and Zhou on graphs that maximize the signless Laplacian spectral radius over all (not necessarily connected) graphs with given numbers of vertices and edges is refuted. The Laplacian spectrum of a (degree) maximal graph is reconsidered.     

The elliptic matrix completion problem

Completions of partial elliptic matrices are studied. Given an undirected graph G, it is shown that every partial elliptic matrix with graph G can be completed to an elliptic matrix if and only if the maximal cliques of G are pairwise disjoint. Further, given a partial elliptic matrix A with undirected graph G, it is proved that if G is chordal and each specified principal submatrix defined by a pair of intersecting maximal cliques is nonsingular, then A can be completed to an elliptic matrix. Conversely, if G is nonchordal or if the regularity condition is relaxed, it is shown that there exist partial elliptic matrices which are not completable to an elliptic matrix. In the process we obtain several results concerning chordal graphs that may be of independent interest.     

Extremal numbers of positive entries of imprimitive nonnegative matrices

We determine the maximum and minimum numbers of positive entries of imprimitive nonnegative matrices with a given imprimitivity index. One application of the results is to estimate the imprimitivity index by the number of positive entries. The proofs involve the study of a cyclic quadratic form. This completes a research initiated by Lewin in 1990.