The existence of left eigenvalues for quaternionic matrix

In this work, an algebraic method to prove the existence of left eigenvalues for the quaternionic matrix is investigated. The left eigenvalues of a [Formula: see text] quaternionic matrix can be derived by solving the zeros of a general quaternionic polynomial of degree [Formula: see text]. Using the Study’s determinant, it can be found by solving the zeros of quaternionic polynomials of degree at most [Formula: see text] or of rational functions.

Bounds for the right spectral radius of quaternionicmatrices

UDC 517.5 In this paper we present bounds for the sum of the moduli of right eigenvalues of a quaternionic matrix. As a consequence, we obtain bounds for the right spectral radius of a quaternionic matrix. We also present a minimal ball in 4D spaces which contains all the Gersgorin balls of a quaternionic matrix. As an application, we introduce the estimation for the right ˇ eigenvalues of quaternionic matrices in the minimal ball. Finally, we suggest some numerical examples to illustrate of our results.

Localization theorems for matrices and bounds for the zeros of polynomials over quaternion division algebra

In this paper, we derive Ostrowski and Brauer type theorems for the left and right eigenvalues of a quaternionic matrix. Generalizations of Gerschgorin type theorems are discussed for the left and the right eigenvalues of a quaternionic matrix. After that, a sufficient condition for the stability of a quaternionic matrix is given that generalizes the stability condition for a complex matrix. Finally, a characterization of bounds is derived for the zeros of quaternionic polynomials.

Cayley-Hamilton theorem for left eigenvalues of 3 × 3 quaternionic matrices

We prove that any quaternionic matrix of order n ≤3 admits a characteristic function, whose roots are the left eigenvalues, that satisfes Cayley-Hamilton theorem.