ALPS: Matrices with nonpositive off-diagonal entries

The purpose of this paper is to characterize and interrelate various degrees of stability and semipositivity for real square matrices having nonpositive off-diagonal entries. The major classes considered are the sets of diagonally stable, stable, and semipositive matrices, denoted respectively by $\text{A}$, $\text{L}$, and $\text{S}$. The conditions defining these classes are weakened, and the resulting classes are examined. Their relationship to the classes of real matrices $\text{P}$ and $\text{P}$₀, whose off-diagonal entries are nonpositive and whose principal minors are respectively all positive and all nonnegative, is also included.     

Extreme points in sets of positive linear maps on B(H)

Three main results are obtained: (1) If $\text{D}$ is an atomic maximal Abelian subalgebra of $\text{B}$($\text{H}$), $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto $\text{D}$ and h is a complex homomorphism on $\text{D}$, then h ° $\text{P}$ is a pure state on $\text{B}$($\text{H}$). (2) If {P_n} is a sequence of mutually orthogonal projections with rank(P_n) = n and ∑ P_n = I, $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto {P_n}″ given by P(T)=∑trace_n(T)P_n and h is a homomorphism on {P_n}″ such that h(P_n) = 0 for all n then h ° $\text{P}$ induces a type II_∞ factor representation of the Calkin algebra. (3) If $\text{M}$ is a nonatomic maximal Abelian subalgebra of $\text{B}$($\text{H}$) then there is an atomic maximal Abelian subalgebra $\text{D}$ of $\text{B}$($\text{H}$) and a large family {Φ_α} of ¹-homomorphisms from $\text{D}$ onto $\text{M}$ such that for each α, Φ_α ° $\text{P}$ is an extreme point in the set of projections from $\text{B}$($\text{H}$) onto $\text{M}$. (Here $\text{P}$ denotes the projection of $\text{B}$($\text{H}$) onto $\text{D}$.)     

The covering problem for the fields Q(−1)12 and Q(−3)12

Let k be a positive square free integer, $\text{N(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ the ring of algebraic integers in $\text{Q(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and S the unit sphere in C_n, complex n-space. If A₁,…, A_n are n linearly independent points of C_n then L = {u₁Au + … + u_nA_n} with $\text{u}{}_{\mathrm{r}}\text{∈ N(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is called a k-lattice. The determinant of L is denoted by d(L). If L is a covering lattice for S, then $\text{θ(S, L) =}\text{V(S)}\text{d(L)}$ is the covering density. L is called locally (absolutely) extreme if θ(S, L) is a local (absolute) minimum. In this paper we determine unique classes of extreme lattices for k = 1 and k = 3.     

On two classes of matrices with positive diagonal solutions to the Lyapunov equation

We characterize real indecomposable quasi-Jacobi matrices of class $\text{D}$, i.e., those which satisfy the Lyapunov equation PA + A′P = ?Q with P diagonal and both P and Q positive definite. The subclass $\text{D}$₂ (of class $\text{D}$) when also Q is diagonal is also characterized in the case of general indecomposable real matrices.     

Lifting commuting pairs of C1 algebras

Given a commuting pair $\text{A}$₁, $\text{A}$₂ of abelian $\text{C}{}^1$ subalgebras of the Calkin algebra, we look for a commuting pair $\text{B}$₁,$\text{B}$₂ of $\text{C}{}^1$ subalgebras of $\text{B(H)}$ which project onto $\text{A}$₁ and $\text{A}$₂. We do not insist that $\text{B}$_i, be abelian, so $\text{B}$_i, may contain nontrivial compact operators. If X is the joint spectrum σ($\text{A}$₁, $\text{A}$₂), it is shown that the existence of a pair $\text{B}$₁, $\text{B}$₂ depends only on the element τ in Ext(X) determined by $\text{A}$₁, $\text{A}$₂. The set L(X) of those τ in Ext(X) which “lift” in this sense is shown to be a subgroup of Ext(X) when Ext(X) is Hausdorff, and also when $\text{A}$_i are singly generated. In this latter case, L(X) can be explicitly calculated for large classes of joint spectra. These results are applied to lift certain pairs of commuting elements of the Calkin algebra to pairs of commuting operators.     

Continuity properties of the averaging projection onto the set of Hankel matrices

Some continuity properties of the averaging projection $\text{P}$ onto the set of Hankel matrices are investigated. It is proved that this projection is of weak type (1, 1) which means that for any nuclear operator T the s-numbers of $\text{P}$T satisfy S_n($\text{P}$T) ? const(n + 1). As a consequence it is obtained that $\text{P}$ maps the Matsaev ideal $\text{G}$_ω = {T:∑_n?0S_n(T)(2n + 1)^?1 < ∞} into the set of compact operators.     

Dimension of the crown Skn

In 1941, Dushnik and Miller introduced the concept of the dimension of a poset (X, P) as the minimum number of linear extensions of P whose intersection is exactly P. Although Dilworth has given a formula for the dimension of distributive lattices, the general problem of determining the dimension of a poset is quite difficult. An equally difficult problem is to classify those posets which are dimension irreducible, i.e., those posets for which the removal of any point lowers the dimension. In this paper, we construct for each n≥3, k≥0, a poset, called a crown and denoted S^k_n, for which the dimension is given by the formula $\text{2?(n+k)}\text{(k+2)}$. Furthermore, for each t≥3, we show that there are infinitely many crowns which are irreducible and have dimension t. We then demonstrate a method of combining a collection of irreducible crowns to form an irreducible poset whose dimension is the sum of the crowns in the collection. Finally, we construct some infinite crowns possessing combinatorial properties similar to finite crowns.     

Structure of H-classes in the semigroup of nonnegative matrices

This paper investigates the structure of the $\text{H}$-classes in the semigroup N_n of nonnegative matrices. We obtain two sets of equivalent conditions for any two matrices A,B to satisfy $\text{A}\text{H}\text{B in N}{}_{\mathrm{n}}$. We establish a one-to-one and onto correspondence between the $\text{H}$-class $\text{H}$_A and the group W_A0 of the greatest cone independent submatrix A₀ of A. We find W_A0 can be made up from the groups of the connective submatrices of A₀.     

A property of matrices with positive determinants

Let $\text{P}$ be the set of all n × n real matrices which have a positive determinant. We show here that at least 2^{n ? 1} matrices are needed to “see” each matrix in $\text{P}$. Also, any finite subset of $\text{P}$ can be “seen” from a class of at most 2^{n ? 1} matrices in $\text{P}$.     

Scalar polynomial functions on the n×n matrices over a finite field

Let F=GF(q) denote the finite field of order q, and let $\text{?(x)?F[x]}$. Then f(x) defines, via substitution, a function from F_n×n, the n×n matrices over F, to itself. Any function $\text{?:F}{}_{\mathrm{n\times n}}\text{→ F}{}_{\mathrm{n\times n}}$ which can be represented by a polynomialf(x)?F[x] is called a scalar polynomial function on F_n×n. After first determining the number of scalar polynomial functions on F_n×n, the authors find necessary and sufficient conditions on a polynomial $\text{?(x) ? F[x]}$ in order that it defines a permutation of (i) $\text{D}$_n, the diagonalizable matrices in F_n×n, (ii)$\text{R}$_n, the matrices in F_n×n all of whose roots are in F, and (iii) the matric ring F_n×n itself. The results for (i) and (ii) are valid for an arbitrary field F.     

Generalized inverse-positivity and splittings of M-matrices

The concepts of matrix monotonicity, generalized inverse-positivity and splittings are investigated and are used to characterize the class of all M-matrices A, extending the well-known property that A^?1?0 whenever A is nonsingular. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. It is shown how the nonnegativity of a generalized left inverse of A plays a fundamental role in such characterizations, thereby extending recent work by one of the authors, by Meyer and Stadelmaier and by Rothblum. In addition, new characterizations are provided for the class of M-matrices with “property c”; that is, matrices A having a representation A=sI?B, s>0, B?0, where the powers of $\text{(}\text{1}\text{s}\text{)B}$ converge. Applications of these results to the study of iterative methods for solving arbitrary systems of linear equations are given elsewhere.     

Minimum number of edges in graphs that are both P2- and Pi-connected

A simple graph with n vertices is called P_i-connected if any two distinct vertices are connected by an elementary path of length i. In this paper, lower bounds of the number of edges in graphs that are both P₂- and P_i-connected are obtained. Namely if $\text{i?}\text{1}\text{2}\text{(n+1)}$, then |E(G)|?((4i?5)/(2i?2))(n?1), and if $\text{i >}\text{1}\text{2}\text{(n+ 1)}$, then |E(G)|?2(n?1) apart from one exeptional graph. Furthermore, extremal graphs are determined in the former.     

On the completely positive and positive-semidefinite-preserving cones

The cone CP_n,q of completely positive linear transformations from $\text{M}{}_{\mathrm{n}}\text{(}\text{C}\text{)=}\text{M}{}_{\mathrm{n}}$ to $\text{M}$_q is shown to be isometrically isomorphic to $\text{P}$_nq, the cone of nq by nq positive semidefinite matrices. Generalizations of scalar and matrix results to $\text{CP}{}_{\mathrm{n,\; q}}\text{?}\text{HP}{}_{\mathrm{n,\; q}}\text{?}\text{L}\text{(}\text{M}{}_{\mathrm{n}}\text{,}\text{M}{}_{\mathrm{q}}\text{)}$ (where HP_n,q represents the hermitian-preserving linear transformations) are discussed. Relationships among the completely positives, the set of positive semidefinite preservers $\text{π(}\text{P}{}_{\mathrm{n}}\text{)}$, and its dual $\text{π(}\text{M}{}_{\mathrm{n}}\text{)}{}^1$ are given. Left and right facial ideals of CP are characterized. Properties of the joint angular field of values of a finite sequence of hermitian matrices H₁,…, H_m are studied, leading to a characterization of $\text{π(}\text{P}{}_{\mathrm{q}}\text{,}\text{P}{}_{\mathrm{n}}\text{)}$.     

Tabloïdes

A tabloïde is composed of two finite sets, E (the set of the rows) and F (the set of the columns), and of a function f: $\text{P}$(E) × $\text{P}$(F) → $\text{N}$, which is a Whitney's rank in its two variables. There are tabloids associated to matrices, to bipartite graphs and to ordinary graphs respectively in relation to linear rank, matchings and gammoids. Some of the properties of matrices can be generalized to tabloids (transposition, direct sum, inverse, product…). The properties of bipartite graphs which consists in “transmission” of matroids by matchings is used to define a class of tabloids (which strictly contains those which are associated to matrices). Finally, the problem of the representation of a tabloid on a field is studied.     

Author index to volume 20

This paper considers the conjecture that given a real nonsingular matrix A, there exist a real diagonal matrix Λ with $\text{¦λ}{}_{\mathrm{ii}}\text{λ = 1}$ and a permutation matrix P such that (ΛPA) is positive stable. The conjecture is shown to be true for matrices of order 3 or less and may not be true for higher order matrices. A counterexample is presented in terms of a matrix of order 65. In showing this, an interesting matrix M_l of order 2^l = 64, which satisfies the matrix equation 2^l-1(M_l + M^T_l), has been used. The stability analysis is done by first decomposing the nonsingular matrix into its polar form. Some interesting results are presented in the study of eigenvalues of a product of orthogonal matrices. A simple function is derived in terms of these orthogonal matrices, which traces a hysteresis loop.     

On maximal weights of Hadamard matrices

Let $\text{Ω}{}_{\mathrm{n}}$ denote the set of all Hadamard matrices of order n. For $\text{H ? Ω}{}_{\mathrm{n}}$, define the weight of H to be the number of 1's in H and is denoted by w(H). For a subset $\text{Γ ? Ω}{}_{\mathrm{n}}$, define the maximal weight of Γ as w(Γ) = max{w(H) | H?Γ}. Two Hadamard matrices are equivalent if one of them can be transformed to the other by permutation and negation of rows and columns, and the equivalence class containing H is denoted by [H]. In this paper, we shall derive lower bounds for w([H]), which are best possible for n ? 20. We shall also determine the exact value of $\text{w(Ω}{}_{32}\text{)}$.     

Nonwastefulness of interior iterative procedures

An algorithm parametrized by sequences of matrices {A_i} and {B_i} is presented. The concept of wastefulness of the algorithm on a class $\text{b}$ of constrained optimization problems is introduced. Then it is proved that the algorithm is nonwasteful on $\text{b}$ if and only if the matrices A_i are positive multiples of the identity matrix and the matrices B_i are positive semidefinite.     

Classification of certain pairs of operators (P, Q) satisfying [P, Q] = −i Id

In a separable Hilbert space a certain class of pairs of operators (P, Q) satisfying the Born-Heisenberg commutation relation [P, Q] = ?i Id on a dense domain Ω is investigated. This class is essentially defined by requiring Q bounded and self-adjoint, P symmetric and $\text{PΩ ? Ω, QΩ ? Ω}$. We show that Q is absolutely continuous and that P can be thought of as a first order differential operator. The class considered contains the pair “angle ?” and “angular momentum L_z.” It is expected that the methods of this paper can be applied to more general classes of operators (P, Q) including the Schrödinger case.