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This paper concerns test matrices for numerical linear algebra using an error-free transformation of floating-point arithmetic. For specified eigenvalues given by a user, we propose methods of generating a matrix whose eigenvalues are exactly known based on, for example, Schur or Jordan normal form and a block diagonal form. It is also possible to produce a real matrix with specified complex eigenvalues. Such test matrices with exactly known eigenvalues are useful for numerical algorithms in checking the accuracy of computed results. In particular, exact errors of eigenvalues can be monitored. To generate test matrices, we first propose an error-free transformation for the product of three matrices YSX. We approximate S by S ′ to compute Y S ′ X without a rounding error. Next, the error-free transformation is applied to the generation of test matrices with exactly known eigenvalues. Note that the exactly known eigenvalues of the constructed matrix may differ from the anticipated given eigenvalues. Finally, numerical examples are introduced in checking the accuracy of numerical computations for symmetric and unsymmetric eigenvalue problems. [ABSTRACT FROM AUTHOR]

For general complex or real 1-parameter matrix flows A(t)n, n and for static matrices A ∈ ℂ n , n alike, this paper considers ways to decompose matrix flows and single matrices globally via one constant matrix similarity Cn, n as A(t) = C− 1 ⋅ diag(A1(t),...,Aℓ(t)) ⋅ C or A = C− 1 ⋅diag(A1,...,Aℓ) ⋅ C with each diagonal block Ak(t) or Ak square and their number ℓ exceeding 1 if this is possible. The theory behind our proposed algorithm is elementary and uses the concept of invariant subspaces for the MATLAB eig computed 'eigenvectors' of one associated flow matrix B(ta) to find the coarsest simultaneous block structure for all flow matrices B(tb). The method works efficiently for all time-varying matrix flows A(t), be they real or complex, normal, with Jordan structures or repeated eigenvalues, differentiable, continuous, or discontinuous in t, and likewise for all fixed entry matrices A. Our intended aim is to discover unitarily diagonal-block decomposable flows as they originate in real-time from sensor given data for time-varying matrix problems that are unitarily invariant. Then, the complexities of their numerical treatments decrease by adopting 'divide and conquer' methods for their diagonal blocks. In the process, we discover and study k-normal fixed entry matrix classes that can be decomposed under unitary similarities into various k-dimensional block-diagonal forms. [ABSTRACT FROM AUTHOR]

Differentiation matrices are in wide use in numerical algorithms, although usually studied in an ad hoc manner. We collect here in this review paper elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial bases and non-degree-graded bases such as Bernstein bases and Lagrange and Hermite interpolational bases. We give new explicit formulations, and new explicit formulations for the pseudo-inverses which help to understand antidifferentiation, of many of these matrices. We also give the unique Jordan form for these (nilpotent) matrices and a new unified formula for the transformation matrix. [ABSTRACT FROM AUTHOR]

If the cosquares of complex matrices and are similar and there is a unimodular number in the spectrum of the cosquares, then and are not necessarily congruent. Assume that such an eigenvalue is unique. In this case, so far, one could verify the congruence of and by using a rational algorithm only in two situations: (1) the eigenvalue is simple or semi-simple; (2)there is only one Jordan block associated with in the Jordan form of the cosquares. We propose a rational algorithm for checking congruence in the case where two Jordan blocks of the same order are associated with . [ABSTRACT FROM AUTHOR]

In this paper, we study the noncommutative functional equation h (λ z) - λ h (z) = g (z) , z ∈ C and we give a new perspective from this equation to obtain a completely dynamical classification for complex matrices. Coarsely speaking, there are four different types: 0 , 1 2 , 2 and e i 2 π θ with θ ∈ [ 0 , 1 2 ] for diagonal matrices and Jordan matrices, respectively. Moreover, we obtain that for a complex matrix A, if its eigenvalues are 0 < | λ i | ≠ 1 , where 1 ≤ i ≤ rank (A) , then A is topologically conjugate to a diagonal matrix on C rank (A) . [ABSTRACT FROM AUTHOR]

For a class of second-order cone nonlinear complementarity problems, abbreviated as SOCNCPs, we establish the modulus-based matrix splitting relaxation iteration methods, which are obtained by reformulating equivalently SOCNCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The global convergence theorems are given under suitable choices of the involved splitting matrix and parameter matrices. When the splitting matrix is symmetric positive definite, the strategy choice of the parameters is discussed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SOCNCPs. [ABSTRACT FROM AUTHOR]

We study the Schwarz problem for J-analytic vector-valued functions in an ellipse with a square matrix J admitting a nondiagonal Jordan form. We obtain conditions on the ellipse and matrix J necessary and sufficient for the existence and uniqueness of a solution to the Schwarz problem with an arbitrary boundary function of Hölder class. Under certain conditions on the matrix J, we show that the homogeneous Schwarz problem in an ellipse has a solution in the form of a vector polynomial of an arbitrary degree. [ABSTRACT FROM AUTHOR]

We study the Schwarz problem for J-analytic functions in the case where the Jordan basis Q of the matrix J has real vectors and the Jordan form J1 of J contains Jordan (2×2)-cells, whereas the order of real columns of the matrix Q depends on the structure of J1. We prove the existence and uniqueness of a solution to the Schwarz problem in Hölder classes. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

The problem of the similarity of integer matrices with single eigenvalue over the ring of integers is considered. A criterion for a matrix to be similar to a Jordan block is obtained. In addition, a similarity criterion for matrices whose minimal polynomial has degree 2 is obtained. [ABSTRACT FROM AUTHOR]