Raphael Loewy

Graphs whose minimal rank is two

Let F be a field, G =( V, E) be an undirected graph on n vertices, and let S(F, G) be the set of all symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F, G )co nsists of the symmetric irreducible tridiagonal matrices. Let mr(F, G) be the minimum rank over all matrices in S(F, G). Then mr(F, G) = 1 if and only if G is the union of a clique with at least 2 vertices and an independent set. If F is an infinite field such that charF � , then mr(F, G) ≤ 2i f and only if the complement of G is the join of a clique and a graph that is the union of at most two cliques and any number of complete bipartite graphs. A similar result is obtained in the case that F is an infinite field with char F = 2. Furthermore, in each case, such graphs are characterized as those for which 6 specific graphs do not occur as induced subgraphs. The number of forbidden subgraphs is reduced to 4 if the graph is connected. Finally, similar criteria is obtained for the minimum rank of a Hermitian matrix to be less than or equal to two. The complement is the join of a clique and a graph that is the union of any number of cliques and any number of complete bipartite graphs. The number of forbidden subgraphs is now 5, or in the connected case, 3.

On vibrations of heterogeneous orthotropic cylindrical shells.

Abstract : A refined Love-type theory of motion is established for orthotropic composite cylindrical shells. An extensional-rotational dynamic coupling effect is shown to exist, expressed by R sub 1 inertia terms. An extended version of the theory is formulated to account for dynamic stability problems involving time-dependent and non-conservative forces. The frequency spectra of free natural vibrations are investigated for numerous layered shells, using Love and Donnell-type theories, including the effects of R sub 1 terms. Heterogeneity is found to considerably affect the results for the natural frequencies; for certain shells produced of a fixed amount of materials, differing only in their arrangement, a suitable composition raises the lowest frequency by a factor of 1.50. A study of the error involved in a Donnell-type theory is carried out. For length-to-radius ratios of about 5 the resulting first lowest frequency may be higher by a factor of 1.10 than the one given by the present Love-type theory. However, when higher frequencies are considered this factor may go down to 0.66. These deviations are, in several instances, associated with different predictions of the corresponding lowest characteristic mode shapes. Higher errors, strongly depending on shell heterogeneity, are noted as the length-to-radius ratios increase beyond 5.

The real and the symmetric nonnegative inverse eigenvalue problems are different

We show that there exist real numbers Al, A2 ... A, that occur as the eigenvalues of an entry-wise nonnegative n-by-n matrix but do not occur as the eigenvalues of a symmetric nonnegative n-by-n matrix. This solves a problem posed by Boyle and Handelman, Hershkowitz, and others. In the process, recent work by Boyle and Handelman that solves the nonnegative inverse eigenvalue problem by appending 0s to given spectral data is refined.

Axisymmetric Vibrations of Isotropic Composite Circular Plates

A Kirchhoff‐type theory is established for axisymmetric motions of heterogeneous isotropic circular plates. It is shown that a coupled extensional‐flexural inertia term exists, in addition to the classical extensional and rotatory inertia terms. An analogy is found between the composite plate problem and the vibrations of homogeneous shallow spherical shells. The obtained sixth‐order system of equations is solved in closed form in terms of Bessel functions, with an argument determined from a characteristic cubic equation. A transcendental frequency equation is then derived for a circular composite plate with clamped edge conditions. Numerous examples are studied, showing the significant effect of plate heterogeneity on its vibrational response. Possibility of composite systems to transcend the frequencies of the individual constituents is clearly indicated by the theoretical results and checked experimentally.

The graphs for which the maximum multiplicity of an eigenvalue is two

Characterized are all simple undirected graphs G such that any real symmetric matrix that has graph G has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general fields), but only certain partial 2-trees guarantee maximum multiplicity 2. Among partial linear 2-trees, they are only those whose vertices can be covered by two ‘parallel’ induced paths. The remaining graphs that guarantee maximum multiplicity 2 are composed of certain identified families of ‘exceptional’ partial 2-trees that are not linear.

CP rank of completely positive matrices of order 5

Abstract J.H. Drew et al. [Linear and Multilinear Algebra 37 (1994) 304] conjectured that for n ⩾4, the completely positive (CP) rank of every n × n completely positive matrix is at most [ n 2 /4]. In this paper we prove that the CP rank of a 5×5 completely positive matrix which has at least one zero entry is at most 6, thus providing new supporting evidence for the conjecture.

Graphs whose minimal rank is two : the finite fields case

Let F be a finite field, G =( V, E) be an undirected graph on n vertices, and let S(F, G) be the set of all symmetric n × n matrices over F whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G.L et mr(F, G) be the minimum rank of all matrices in S(F, G). If F is a finite field with p t elements, p � , it is shown that mr(F, G) ≤ 2i f and only if the complement of G is the join of a complete graph with either the union of at most (p t +1)/2 nonempty complete bipartite graphs or the union of at most two nonempty complete graphs and of at most (p t − 1)/2 nonempty complete bipartite graphs. These graphs are also characterized as those for which 9 specific graphs do not occur as induced subgraphs. If F is a finite field with 2t elements, then mr(F, G) ≤ 2 if and only if the complement of G is the join of a complete graph with either the union of at most 2t + 1 nonempty complete graphs or the union of at most one nonempty complete graph and of at most 2 t−1 nonempty complete bipartite graphs. A list of subgraphs that do not occur as induced subgraphs is provided for this case as well.

Linear transformations which preserve or decrease rank

Abstract We describe some results concerning a linear transformation on a space V of matrices, which is rank preserving or rank nonincreasing on a certain subset of V .

Rank preservers on spaces of symmetric matrices

LetSn (F) denote the set of all n × n symmetric matrices over the field F. Let k be a positive integer such that k ≤ n. Alinear operator T on Sn (F) is said to be a rank-k preserver provided that it maps the set of all rank k matrices into itself. We show here that if k is even and F is algebraically closed of characteristic ≠ 2, then any such T must be a congruence map. The corresponding result for k odd has already been established. Now suppose that r is a positive integer such that 2r≤ n. We also consider a linear operator T on the space of n × n hermitian matrices, which maps the set of matrices whose inertia is (r,r,n – 2r) into itself. In the real symmetric case we show that if n ≥ 4r any such T must be a congruence map, possibly followed by negation. This proves the Johnson-Pierce conjecture for this inertia class, improving known results. An analogous result is obtained in the hermitian case.

Spaces of symmetric matrices of bounded rank

Abstract Let I n denote the space of all n×n symmetric matrices over a field F . Let t be a positive integer such that t . A subspace W of I n ( F ) is said to be a t -subspace provided that the rank of every matrix in W is bounded by t . Meshulam showed, under the assumption | F |⩾ n + 1, that the maximal dimension of a t -subspace of I n ( F ) is given by max t+1 2 , k+1 2 +k(n−k) if t2k, max t+1 2 , k+1 2 +k(n−k)+1 if t2k+1 Provided that we also assume char F ≠2, we show here that any t -subspace of I n ( F ) of maximal dimension is congruent to W 1 (n,t){A∈ I n (F):a ij 0 if i>t or j>t},