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Zeros of unilateral quaternionic polynomials. (English) Zbl 1151.15303
Summary: The purpose of this paper is to show how the problem of finding the zeros of unilateral $$n$$-order quaternionic polynomials can be solved by determining the eigenvectors of the corresponding companion matrix. This approach, probably superfluous in the case of quadratic equations for which a closed formula can be given, becomes truly useful for (unilateral) $$n$$-order polynomials.
To understand the strength of this method, it is compared with the algorithm of I. Niven [Am. Math. Mon. 49, 386–388 (1942; Zbl 0061.01407)] and it is shown where this (full) matrix approach improves previous methods based on the use of the Niven algorithm. For convenience of the readers, some examples of second and third order unilateral quaternionic polynomials are explicitly solved. The leading idea of the practical solution method proposed in this work can be summarized in the following three steps: translating the quaternionic polynomial in the eigenvalue problem for its companion matrix, finding its eigenvectors, and, finally, giving the quaternionic solution of the unilateral polynomial in terms of the components of such eigenvectors. A brief discussion on bilateral quaternionic quadratic equations is also presented.

##### MSC:
 15B33 Matrices over special rings (quaternions, finite fields, etc.) 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 12Y05 Computational aspects of field theory and polynomials (MSC2010) 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 15A18 Eigenvalues, singular values, and eigenvectors
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