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Abstract Whether the n × n completely positive cone for n ≥ 5 is facially exposed is an open problem raised by Berman et al. in 2015. In this paper, we give a class of non-exposed faces of the 5 × 5 completely positive cone, which implies that the 5 × 5 completely positive cone is not facially exposed. By embedding the 5 × 5 completely positive cone into the n × n completely positive cone for n > 5 , we further show that the n × n completely positive cone is not facially exposed for n > 5. [ABSTRACT FROM AUTHOR]

It is known that every complex square matrix with nonnegative determinant is the product of positive semi-definite matrices. There are characterizations of matrices that require two or five positive semi-definite matrices in the product. However, the characterizations of matrices that require three or four positive semi-definite matrices in the product are lacking. In this paper, we give a complete characterization of these two types of matrices. With these results, we give an algorithm to determine whether a square matrix can be expressed as the product of k positive semi-definite matrices but not fewer, for k = 1 , 2 , 3 , 4 , 5 . [ABSTRACT FROM AUTHOR]

Abstract: Following an idea of spectral graph theory, we give several characterizations of finite one-peak posets I with almost P-critical quadratic Tits form by applying linear algebra methods, combinatorial algorithms, an old idea of S.A. Ovsienko and recent results by V.M. Bondarenko and M.V. Stepochkina. Our study is mainly motivated by an important application of the quadratic Tits form of a poset in constructing linear algebra invariants that measure a geometric complexity of Nazarova–Roiter matrix problems over a field K. In particular, our study is inspired by a well-known result of Ju.A. Drozd explained in Introduction. One of our main results asserts that, given a finite poset such that ⁎ is its unique maximal element and is almost P-critical, we have: [(i)] the form is non-negative and the subgroup of is generated by a vector , with , [(ii)] and the Coxeter polynomial coincides with the Coxeter polynomial of a uniquely determined Euclidean diagram , [(iii)] the -bilinear Tits form of I is -equivalent to the Gram -bilinear form of the Euclidean diagram , [(iv)] there is a -invertible matrix C such that and , where is the Tits matrix of I, and [(v)] I is one of the posets listed in the paper. In particular, we get in an alternative way the list of P-critical posets T (in the form ) presented by V.M. Bondarenko and M.V. Stepochkina [6], and we determine their Coxeter–Euclidean types. There are 115 one-peak posets , with P-critical, and 17 exceptional posets , with almost P-critical, but not P-critical. Our lists are obtained by using computer algebra tools; mainly symbolic computation in Maple and numeric computation in C#. [Copyright &y& Elsevier]

Abstract: In this paper we investigate structured backward errors for three kinds of generalized saddle point systems where the matrix is not symmetric and its block is not zero and has perturbations. Computable formula of backward errors are derived. The expressions are useful for testing the stability of practical algorithms. [Copyright &y& Elsevier]

Abstract: In a Coalitional Resource Game (CRG for brief), agents form coalitions to pool their resources in order to achieve certain goals, requiring the expenditure of these resources. A particular coalition is said to be successful, if the common resources of its members enable to achieve a set of goals that satisfies all members of the coalition. It is known that when resources are consumable it is NP-complete to decide whether a given coalition is successful. In this paper, we show a connection of CRGs with sharable resources and max–min linear systems of inequalities. This correspondence leads to polynomial algorithms for checking whether a given CRG admits a successful coalition and for several other problems whose counterparts for CRGs with consumable resources are hard. On the other hand, we prove that some problems concerning the structure of successful coalitions are hard also in the case of sharable resources. [Copyright &y& Elsevier]

Abstract: The tridiagonal Birkhoff polytope, , is the set of real square matrices with nonnegative entries and all rows and columns sums equal to 1 that are tridiagonal. This polytope arises in many problems of enumerative combinatorics, statistics, combinatorial optimization, etc. In this paper, for a given a -face of , we determine the number of faces of lower dimension that are contained in it and we discuss its nature. In fact, a 2-face of is a triangle or a quadrilateral and the cells can only be tetrahedrons, pentahedrons or hexahedrons. [Copyright &y& Elsevier]

Abstract: As is well known, a rank- matrix can be recovered from a cross of linearly independent columns and rows, and an arbitrary matrix can be interpolated on the cross entries. Other entries by this cross or pseudo-skeleton approximation are given with errors depending on the closeness of the matrix to a rank- matrix and as well on the choice of cross. In this paper we extend this construction to -dimensional arrays (tensors) and suggest a new interpolation formula in which a -dimensional array is interpolated on the entries of some TT-cross (tensor train-cross). The total number of entries and the complexity of our interpolation algorithm depend on linearly, so the approach does not suffer from the curse of dimensionality. We also propose a TT-cross method for computation of -dimensional integrals and apply it to some examples with dimensionality in the range from up to and the relative accuracy of order . In all constructions we capitalize on the new tensor decomposition in the form of tensor trains (TT-decomposition). [Copyright &y& Elsevier]

Abstract: In this paper, indefinite linear systems with linear constraints are considered. We present a special decomposition that makes use of the LQ decomposition, and retains the constraints in the factors. The resulting decomposition is of a structure similar to that obtained using the Bunch–Kaufman–Parlett algorithm. The decomposition can be used in a direct solution algorithm for indefinite systems, but it can also be used to construct effective preconditioners. Combinations of the latter with conjugate gradient type methods have been demonstrated to be very useful. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the problem of maximizing the coupling between the isometric projections of two square matrices of dimensions m and n. This coupling is defined as an inner product between the matrices. This is a non-convex optimization problem with isometry constraints on the variables. The optimization set is an equinormed set and we develop a gradient-type algorithm to solve the problem. Numerical experiments and an application to graph matching are also presented. [Copyright &y& Elsevier]

Abstract: In this paper, algorithms are developed for the problems of spectral factorization and sum of squares of polynomial matrices with n indeterminates, and a natural interpretation of the tools employed in the algorithms is given using ideas from the theory of lossless and dissipative systems. These algorithms are based on the calculus of 2n-variable polynomial matrices and their associated quadratic differential forms, and share the common feature that the problems are lifted from the original n-variable polynomial context to a 2n-variable polynomial context. This allows to reduce the spectral factorization problem and the sum of squares problem to linear matrix inequalities (LMI’s), to the feasibility of a semialgebraic set or to a linear eigenvalue problem. [Copyright &y& Elsevier]

Abstract: This paper presents an algorithm for computing the symmetric singular value decomposition of square Hankel matrices of order n, in contrast with existing SVD algorithms. The algorithm consists of two stages: first, a complex square Hankel matrix is reduced to a complex symmetric tridiagonal matrix using the block Lanczos method in flops; Second, the singular values and singular vectors of the symmetric tridiagonal matrix resulted from the first stage are computed in flops. The singular vector matrix is given in the form of a product of three or two unitary matrices. The performance of our algorithm is demonstrated by comparing it with the SVD subroutines in Matlab and LAPACK. [Copyright &y& Elsevier]

Abstract: This paper describes a generalization of the isometric Arnoldi algorithm and shows that it can be interpreted as a structured form of modified Gram–Schmidt. Given an isometry A, the algorithm efficiently orthogonalizes the columns of a sequence of matrices M j for j ⩾0 (with M −1 =0) for which the columns of M j − AM j−1 are in a fixed finite dimensional subspace for each j ⩾0. The dimension of the subspace is analogous to displacement rank in the generalized Schur algorithm. The algorithm is described in terms of projections and inner products. This is in contrast to orthogonalization methods based on the generalized Schur algorithm, for which Cholesky factorization is central to the computation. Numerical experiments suggest that, relative to a generalized Schur algorithm, the new algorithm improves the numerical orthogonality of the computed orthonormal sequence. [Copyright &y& Elsevier]

The Schur complement has many applications in linear algebra and in numerical analysis. Schur complement identities have been used for implementing some scalar and vector extrapolation methods. In this paper we define some matrix sequence transformations and give two new Schur complement identities that will be used to derive algorithms for implementing these transformations. [Copyright &y& Elsevier]