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Solving linear system of differential equations by Jordan canonical form needs the change in the real field to complex and then retrieve the complex solutions to the real ones. B. Malesevic, D. Todoric, I. Jovovic, and S. Telebakovic suggest that it is more convenient to apply the rational canonical form than the Jordan canonical form. They reduce the linear system D Y = A Y to higher order differential equations f j (D) z = 0 , where the polynomials f 1 , f 2 , ... , f k appear in the rational canonical form of A. ( D = d / d x .) The present paper proves the converse of their theorems and discovers certain dimensionality relations between the solution spaces of the two types of differential equation problems. The approach is pure algebraic and leaves the door open for extensions to formal differential equations. [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]

In this article, a class of upwind schemes is proposed for systems, each of which yields an incomplete set of linearly independent eigenvectors. The theory of Jordan canonical forms is used to complete such sets through the addition of generalized eigenvectors. A modified Burgers' system and its extensions generate δ,δ′, δ″,⋯,δn waves as solutions. The performance of flux difference splitting-based numerical schemes is examined by considering various numerical examples. Since the flux Jacobian matrix of pressureless gas dynamics system also produces an incomplete set of linearly independent eigenvectors, a similar framework is adopted to construct a numerical algorithm for a pressureless gas dynamics system. [ABSTRACT FROM AUTHOR]