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This paper adapts look-ahead and backward finite difference formulas to compute future eigenvectors and eigenvalues of piecewise smooth time-varying symmetric or hermitean matrix flows A (t). It is based on the Zhang Neural Network (ZNN) model for time-varying problems and uses the associated error function E (t) = A (t) V (t) − V (t) D (t) or e i (t) = A (t) v i (t) − λ i (t) v i (t) with the Zhang design stipulation that E ˙ (t) = − η E (t) or e ˙ i (t) = − η e i (t) with η > 0 so that E (t) and e (t) decrease exponentially over time. This leads to a discrete-time differential equation of the form P (t k) z ˙ (t k) = q (t k) for the eigendata vector z (t k) of A (t k). Convergent look-ahead finite difference formulas of varying error orders then allow us to express z (t k + 1) in terms of earlier A and z data. Numerical tests, comparisons and open questions complete the paper. [ABSTRACT FROM AUTHOR]

Over any field F every square matrix A can be factored into the product of two symmetric matrices as A = S 1 ⋅ S 2 with S i = S i T ∈ F n , n and either factor can be chosen nonsingular, as was discovered by Frobenius in 1910. Frobenius' symmetric matrix factorization has been lying almost dormant for a century. The first successful method for computing matrix symmetrizers, i.e., symmetric matrices S such that SA is symmetric, was inspired by an iterative linear systems algorithm of Huang and Nong (2010) in 2013 [29,30] . The resulting iterative algorithm has solved this computational problem over R and C , but at high computational cost. This paper develops and tests another linear equations solver, as well as eigen- and principal vector or Schur Normal Form based algorithms for solving the matrix symmetrizer problem numerically. Four new eigendata based algorithms use, respectively, SVD based principal vector chain constructions, Gram–Schmidt orthogonalization techniques, the Arnoldi method, or the Schur Normal Form of A in their formulations. They are helped by Datta's 1973 method that symmetrizes unreduced Hessenberg matrices directly. The eigendata based methods work well and quickly for generic matrices A and create well conditioned matrix symmetrizers through eigenvector dyad accumulation. But all of the eigen based methods have differing deficiencies with matrices A that have ill-conditioned or complicated eigen structures with nontrivial Jordan normal forms. Our symmetrizer studies for matrices with ill-conditioned eigensystems lead to two open problems of matrix optimization. [ABSTRACT FROM AUTHOR]

Abstract: We use methods of geometric computing combined with hermitean matrix eigenvalue/eigenvector evaluations to find the generalized Crawford number of a pair of hermitean matrices and quickly and to high precision. The classical Crawford number is defined as the minimal distance from zero in to the field of values of the associated complex matrix if zero lies outside the field of values of A. We describe, test, and compare a geometry based MATLAB code for finding for the generalized Crawford number that measures the smallest distance of zero from the boundary of the field of values of A even if zero lies inside. [Copyright &y& Elsevier]

We develop the philosophical and practical background for teaching an elementary Linear Algebra course from the requirements that are particular to the subject. It mixes the inner workings and logic of Linear Algebra and matrices with concepts, hands-on computational schemes, and applications for a satisfying comprehensive teaching and learning experience. [Copyright &y& Elsevier]

It is known that for every real square matrix A there exists a nonsingular real symmetric matrix S such thatSA=A′S,where A′ denotes the transpose of A.Using the notion of an M-matrix we derive a criterion for A to satisfy the above equality with a diagonal S of signature k. Such a matrix A will be called Dk-symmetrizable and the paper presents some results on this concept. In particular we show that a Dk-symmetrizable matrix shares many properties with a real symmetric matrix and that any real matrix A, up to an orthogonal similarity, is Dk-symmetrizable for some k. [Copyright &y& Elsevier]

Abstract: We study the completion problem for an orthogonal projector A of from knowledge of one or several of its columns, as well as the related matrix inverse problem of finding an orthogonal projector A with AX =Y for two given n by k matrices X and Y. Using geometric properties of orthogonal projections and of the singular value decomposition, we derive theoretical and numerical solutions for these problems. We give algorithms and test their MATLAB implementations. [Copyright &y& Elsevier]