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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: 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: 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: 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]