Search

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]

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]