Linear preservers of row-dense matrices

Let M_m,n be the set of all m × n real matrices. A matrix A ∈ M_m,n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T: M_m,n → M_m,n that preserve or strongly preserve row-dense matrices, i.e., T(A) is row-dense whenever A is row-dense or T(A) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A ∈ M_n,m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear preservers (strong linear preservers) of column-dense matrices is found.     

Globally convergent Jacobi methods for positive definite matrix pairs

The paper derives and investigates the Jacobi methods for the generalized eigenvalue problem A x = λ B x, where A is a symmetric and B is a symmetric positive definite matrix. The methods first “normalize” B to have the unit diagonal and then maintain that property during the iterative process. The global convergence is proved for all such methods. That result is obtained for the large class of generalized serial strategies from Hari and Begovi? Kova? (Trans. Numer. Anal. (ETNA) 47, 107–147, 2017). Preliminary numerical tests confirm a high relative accuracy of some of those methods, provided that both matrices are positive definite and the spectral condition numbers of Δ _A AΔ _A and Δ _B BΔ _B are small, for some nonsingular diagonal matrices Δ _A and Δ _B .     

Two Bilinear (3 × 3)-Matrix Multiplication Algorithms of Complexity 25

As is known, a bilinear algorithm for multiplying 3 × 3 matrices can be constructed by using ordered triples of 3 × 3 matrices A _ρ , B _ρ , C _ρ , \(\rho = \overline {1,r} ,\) where r is the complexity of the algorithm. Algorithms with various symmetries are being extensively studied. This paper presents two algorithms of complexity 25 possessing the following two properties (symmetries): (1) the matricesA₁,B₁, and C1 are identity, (2) if the algorithm involves a tripleA, B, C, then it also involves the triples B, C, A and C, A, B. For example, these properties are inherent in the well-known Strassen algorithm for multiplying 2 × 2 matrices. Many existing (3 × 3)-matrix multiplication algorithms have property (2). Methods for finding new algorithms are proposed. It is shown that the found algorithms are different and new.     

Covering properties and square principles

Covering matrices were used by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom and later by Sharon and Viale to investigate the impact of stationary reflection on the approachability ideal. In the course of this work, they isolated two reflection principles, CP and S, which may hold of covering matrices. In this paper, we continue previous work of the author investigating connections between failures of CP and S and variations on Jensen’s square principle. We prove that, for a regular cardinal λ > ω ₁, assuming large cardinals, □(λ, 2) is consistent with CP(λ, θ) for all θ with θ ⁺ < λ. We demonstrate how to force nice θ-covering matrices for λ which fail to satisfy CP and S. We investigate normal covering matrices, showing that, for a regular uncountable κ, □ _κ implies the existence of a normal ω-covering matrix for κ ⁺ but that cardinal arithmetic imposes limits on the existence of a normal θ-covering matrix for κ ⁺ when θ is uncountable. We introduce the notion of a good point for a covering matrix, in analogy with good points in PCF-theoretic scales. We develop the basic theory of these good points and use this to prove some non-existence results about covering matrices. Finally, we investigate certain increasing sequences of functions which arise from covering matrices and from PCF-theoretic considerations and show that a stationary reflection hypothesis places limits on the behavior of these sequences.     

Hölder rigidity for matrices

It is proved that a (C ₁, C ₂)-Hölder valuation is (2, α)-equivalent to classical valuation on the set of matrices over a skew field and on the set of cubic matrices over a field. These results provide an extension of the Garcia theorem.     

A gradual rank increasing process for matrix completion

The matrix completion problem is easy to state: let A be a given data matrix in which some entries are unknown. Then, it is needed to assign “appropriate values” to these entries. A common way to solve this problem is to compute a rank-k matrix, B _k , that approximates A in a least squares sense. Then, the unknown entries in A attain the values of the corresponding entries in B _k . This raises the question of how to determine a suitable matrix rank. The method proposed in this paper attempts to answer this question. It builds a finite sequence of matrices \(B_{k}, k = 1, 2, \dots \), where B _k is a rank-k matrix that approximates A in a least squares sense. The computational effort is reduced by using B _k-1 as starting point in the computation of B _k . The ability of B _k to serve as substitute for A is measured with two objective functions: a “training” function that measures the distance between the known part of A and the corresponding part of B _k , and a “probe” function that assesses the quality of the imputed entries. Watching the changes in these functions as k increases enables us to find an optimal matrix rank. Numerical experiments illustrate the usefulness of the proposed approach.     

A stability result using the matrix norm to bound the permanent

We prove a stability version of a general result that bounds the permanent of a matrix in terms of its operator norm. More specifically, suppose A is an n × n matrix over C (resp. R), and let P denote the set of n × n matrices over C (resp. R) that can be written as a permutation matrix times a unitary diagonal matrix. Then it is known that the permanent of A satisfies |per(A)| ≤ ||A|| ⁿ ₂ with equality iff A/||A||₂ ∈ P (where ||A||₂ is the operator 2-norm of A). We show a stability version of this result asserting that unless A is very close (in a particular sense) to one of these extremal matrices, its permanent is exponentially smaller (as a function of n) than ||A|| ⁿ ₂. In particular, for any fixed α, β > 0, we show that |per(A)| is exponentially smaller than ||A|| ⁿ ₂ unless all but at most αn rows contain entries of modulus at least ||A||₂(1?β).     

G-matrices,J-orthogonal matrices,and their sign patterns

A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D₁ and D₂ such that A^?T = D₁AD₂, where A^?T denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or ?1. A nonsingular real matrix Q is called J-orthogonal if Q^TJQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided.     

Possible isolation number of a matrix over nonnegative integers

Let ?₊ be the semiring of all nonnegative integers and A an m × n matrix over ?₊. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k×n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of A. For A with isolation number k, we investigate the possible values of the rank of A and the Boolean rank of the support of A. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of m×n matrices whose isolation number is m. That is, those matrices are permutationally equivalent to a matrix A whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.     

The Corona problem on a complete ultrametric algebraically closed field

Let IK be a complete ultrametric algebraically closed field and let A be the Banach IK-algebra of bounded analytic functions in the ”open” unit disk D of IK provided with the Gauss norm. Let Mult(A, ‖. ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Mult _m (A, ‖. ‖) be the subset of the ? ∈ Mult(A, ‖. ‖) whose kernel is amaximal ideal and let Mult ₁(A, ‖. ‖) be the subset of the ? ∈ Mult(A, ‖. ‖) whose kernel is a maximal ideal of the form (x ? a)A with a ∈ D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem the question whether Mult ₁(A, ‖. ‖) is dense in Mult _m (A, ‖. ‖). In a previous paper, it was proved that when IK is spherically complete, the answer is yes. Here we generalize this result to any algebraically closed complete ultrametric field, which particularly applies to ? _p . On the other hand, we also show that the continuous multiplicative seminorms whose kernel are neither a maximal ideal nor the zero ideal, found by Jesus Araujo, also lie in the closure of Mult ₁(A, ‖. ‖), which suggest that Mult ₁(A, ‖. ‖)might be dense in Mult(A, ‖. ‖).     

On Lebesgue Measure of Integral Self-Affine Sets

Let A be an expanding integer n×n matrix and D be a finite subset of ? ⁿ . The self-affine set T=T(A,D) is the unique compact set satisfying the equality \(A(T)=\bigcup_{d\in D}(T+d)\). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈? ⁿ , and the measure of the intersection of self-affine sets T(A,D ₁)∩T(A,D ₂) for different sets D ₁, D ₂?? ⁿ .     

Palindromic widths of nilpotent and wreath products

We prove that the nilpotent product of a set of groups A ₁,…,A _s has finite palindromic width if and only if the palindromic widths of A _i ,i=1,…,s,are finite. We give a new proof that the commutator width of F _n ?K is infinite, where F _n is a free group of rank n≥2 and K is a finite group. This result, combining with a result of Fink [9] gives examples of groups with infinite commutator width but finite palindromic width with respect to some generating set.     

Turán-Type Results for Complete <Emphasis Type="Italic">h</Emphasis>-Partite Graphs in Comparability and Incomparability Graphs

We consider an h-partite version of Dilworth’s theorem with multiple partial orders. Let P be a finite set, and let <₁,...,< _r be partial orders on P. Let G(P, <₁,...,< _r ) be the graph whose vertices are the elements of P, and x, y ∈ P are joined by an edge if x< _i y or y< _i x holds for some 1 ≤ i ≤ r. We show that if the edge density of G(P, <₁, ... , < _r ) is strictly larger than 1 ? 1/(2h ? 2) ^r , then P contains h disjoint sets A ₁, ... , A _h such that A ₁ < _j ... < _j A _h holds for some 1 ≤ j ≤ r, and |A ₁| = ... = |A _h | = Ω(|P|). Also, we show that if the complement of G(P, <) has edge density strictly larger than 1 ? 1/(3h ? 3), then P contains h disjoint sets A ₁, ... , A _h such that the elements of A _i are incomparable with the elements of A _j for 1 ≤ i < j ≤ h, and |A ₁| = ... = |A _h | = |P|^1?o(1). Finally, we prove that if the edge density of the complement of G(P, <₁, <₂) is α, then there are disjoint sets A, B ? P such that any element of A is incomparable with any element of B in both <₁ and <₂, and |A| = |B| > n ^1?γ(α), where γ(α) → 0 as α → 1. We provide a few applications of these results in combinatorial geometry, as well.     

Involutions in Weyl group of type <Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>

Let W be the Weyl group of type F ₄: We explicitly describe a finite set of basic braid I _*-transformations and show that any two reduced I _*-expressions for a given involution in W can be transformed into each other through a series of basic braid I _*-transformations. Our main result extends the earlier work on the Weyl groups of classical types (i.e., A _n , B _n , and D _n ).     

Block partitions of sequences

Given a sequence A = (a ₁, …, a _n ) of real numbers, a block B of A is either a set B = {a _i , a _i+1, …, a _j } where i ≤ j or the empty set. The size b of a block B is the sum of its elements. We show that when each a _i ∈ [0, 1] and k is a positive integer, there is a partition of A into k blocks B ₁, …, B _k with |b _i ?b _j | ≤ 1 for every i, j. We extend this result in several directions.     

A Minimum Principle for Potentials with Application to Chebyshev Constants

For “Riesz-like” kernels K(x,y) = f(|x?y|) on A×A, where A is a compact d-regular set \(A\subset \mathbb {R}^{p}\), we prove a minimum principle for potentials \(U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)\), where μ is a Borel measure supported on A. Setting \(P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)\), the K-polarization of μ, the principle is used to show that if {ν _N } is a sequence of measures on A that converges in the weak-star sense to the measure ν, then P _K (ν _N )→P _K (ν) as \(N\to \infty \). The continuous Chebyshev (polarization) problem concerns maximizing P _K (μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes P _K (μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {ν _N } is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of ν _N as \(N \to \infty \) is a solution to the continuous problem.     

Computational complexity of the original and extended diophantine Frobenius problem

We deduce the law of nonstationary recursion which makes it possible, for given a primitive set A = {a ₁,...,a _k }, k > 2, to construct an algorithm for finding the set of the numbers outside the additive semigroup generated by A. In particular, we obtain a new algorithm for determining the Frobenius numbers g(a ₁,...,a _k ). The computational complexity of these algorithms is estimated in terms of bit operations. We propose a two-stage reduction of the original primitive set to an equivalent primitive set that enables us to improve complexity estimates in the cases when the two-stage reduction leads to a substantial reduction of the order of the initial set.     

Classification of Bott towers by matrix

A criterion for the classification of Bott towers is presented, i.e., two Bott towers B _*(A) and B _*(A′) are isomorphic if and only if the matrices A and A′ are equivalent. The equivalence relation is defined by two operations on matrices. And it is based on the observation that any Bott tower B _*(A) is uniquely determined by its structure matrix A, which is a strictly upper triangular integer matrix. The classification of Bott towers is closely related to the cohomological rigidity problem for both Bott towers and Bott manifolds.     

On the Subspace Projected Approximate Matrix method

We provide a comparative study of the Subspace Projected Approximate Matrix method, abbreviated SPAM, which is a fairly recent iterative method of computing a few eigenvalues of a Hermitian matrix A. It falls in the category of inner-outer iteration methods and aims to reduce the costs of matrix-vector products with A within its inner iteration. This is done by choosing an approximation A ₀ of A, and then, based on both A and A ₀, to define a sequence (A _k ) _k=0 ⁿ of matrices that increasingly better approximate A as the process progresses. Then the matrix A _k is used in the kth inner iteration instead of A.In spite of its main idea being refreshingly new and interesting, SPAM has not yet been studied in detail by the numerical linear algebra community. We would like to change this by explaining the method, and to show that for certain special choices for A ₀, SPAM turns out to be mathematically equivalent to known eigenvalue methods. More sophisticated approximations A ₀ turn SPAM into a boosted version of Lanczos, whereas it can also be interpreted as an attempt to enhance a certain instance of the preconditioned Jacobi-Davidson method.Numerical experiments are performed that are specifically tailored to illustrate certain aspects of SPAM and its variations. For experiments that test the practical performance of SPAM in comparison with other methods, we refer to other sources. The main conclusion is that SPAM provides a natural transition between the Lanczos method and one-step preconditioned Jacobi-Davidson.     

Commutators of trace zero matrices over principal ideal rings

We prove that for every trace zero square matrix A of size at least 3 over a principal ideal ring R, there exist trace zero matrices X, Y over R such that XY ? YX = A. Moreover, we show that X can be taken to be regular mod every maximal ideal of R. This strengthens our earlier result that A is a commutator of two matrices (not necessarily of trace zero), and in addition, the present proof is simpler than the earlier one.