Abstract We address the practical issue that popular computation platforms like Matlab $$^{\copyright }$$ © are unable to perform Jordan normal (canonical) form J and associated transform matrix P on a high-dimension matrix. Given knowledge of eigenvalues eigenvectors n -sq...

In his CAV 2004 paper, Tiwari has proved that, for a class of linear programs over the reals, termination is decidable. And Tiwari has shown that the termination of a linear program P1 whose assignment matrix à is not in the Jordan canonical form is equivalent to that of a linear program J1, whose assignment matrix A is in the Jordan Canonical Form. In most cases, the method of Tiwari provides ...

First order perturbation theory for eigenvalues of arbitrary matrices is systematically developed in all its generality with the aid of the Newton diagram, an elementary geometric construction first proposed by Isaac Newton. In its simplest form, a square matrix A with known Jordan canonical form is linearly perturbed to A(ε) = A+ ε B for an arbitrary perturbation matrix B, and one is intereste...

We describe canonical forms for elements of a classical Lie group of matrices under similarity transformations in the group. Matrices in the associated Lie algebra and Jordan algebra of matrices inherit related forms under these similarity transformations. In general, one cannot achieve diagonal or Schur form, but the form that can be achieved displays the eigenvalues of the matrix. We also dis...

Let A∈Rn×n be an irreducible totally nonnegative matrix with rank r and principal p, that is, all minors of A are nonnegative, is the size largest invertible square submatrix p its submatrix. triple (n,r,p) called realizable if there exists n×n p. In this work we present a method to construct matrices associated prescribed Jordan canonical form corresponding zero eigenvalue.

Let V be a vector space over some field k, and let α : V V be a linear map (an ‘endomorphism of V ’). Given any polynomial p with coefficients in k, there is an endomorphism p(α) of V , and we say that p is an annihilating polynomial for α if p(α) = 0. Our first major goal is to see that for any α, the annihilating polynomials can easily be classified: they’re precisely the multiples of a certa...

Given a finite field F and a positive integer n, we give a procedure to count the n×n matrices with entries in F with all eigenvalues in the field. We give an exact value for any field for values of n up to 4, and prove that for fixed n, as the size of the field increases, the proportion of matrices with all eigenvalues in the field approaches 1/n!. As a corollary, we show that for large fields...

We apply the Born–Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures. In the examples considered, we have that the Weyl formula always preserves t...