On the Quaternionic Quadratic Equation $$\varvec{xax+bx+xc+d=0}$$ x a
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Advances in Applied Clifford Algebras
On the Quaternionic Quadratic Equation xax + bx + xc + d = 0 E. Mac´ıas-Virg´os∗
and M. J. Pereira-S´aez
Abstract. It is shown that the roots of the quaternion quadratic equation xax + bx + xc + d = 0 can be found by computing the eigenvalues of a companion matrix. This allows to completely discuss the number of solutions and to apply some Gerˇsgorin type theorems. Another consequence is the possibility of constructing equations with a given number of solutions. Mathematics Subject Classification. Primary 11R52; Secondary 55M30, 12E15, 65H04. Keywords. Quaternion matrix, Right eigenvalue, Quadratic equation, Companion matrix, Gerˇsgorin ball.
1. Introduction Solving quaternionic equations is an active field of research [3,7,13,22]. For the unilateral quadratic equation ax2 +bx+c = 0, explicit solutions were given for the first time by Huang and So [11] in 2001. Known methods of solution of those quaternionic equations usually lead to consider an equivalent non-linear system of real polynomial equations. Alternatively, when the quaternionic n polynomial is unilateral, that is, of the form i=0 ai xi = 0, its roots can be found by computing the eigenvalues of a so-called companion matrix [6,17, 21]. In this paper we consider the quadratic equation xax + bx + xc + d = 0,
a, b, c, d ∈ H,
a = 0,
(1.1)
that was solved by Au-Yeung in [2], see also [12]. Other authors call this equation an “algebraic Riccati equation” [4,15]. In fact, the polynomial in (1.1) can be reduced to a unilateral one by means of the transformation x = a−1 (y−c), but the solutions are usually given through rather complicated This work was partially supported by MTM2016-78647-P research project from MINECOSpain and FEDER. The first author was also supported by Xunta de Galicia 2015-PG006. ∗ Corresponding
author. 0123456789().: V,-vol
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E. Mac´ıas-Virg´ os, M. J. Pereira-S´ aez
Adv. Appl. Clifford Algebras
formulas. Also, it can be reduced to the form z 2 + pz + q, with p, q ∈ H, (p) = 0 [15, Theorem 3.1]. We shall prove the following result, which is an adaptation to the non-unilateral case of the companion matrix method cited above, and greatly simplifies the discussion and resolution of the equation: Theorem 1.1. The roots of Eq. (1.1) are exactly the quaternions of the form x = a−1 (β − c), where β is a privileged eigenvalue of the quaternionic matrix −b −d A= . (1.2) a c The notion of privileged eigenvalue will be reviewed in Sect. 2, while Theorem 1.1 will be proved in Sect. 3. As a consequence, we establish in Sect. 4 a complete classification of the equations in terms of its number of solutions, that is, one, two or an infinite number, which is based in the Jordan form of the companion matrix (Theorem 4.2). Another application of Theorem 1.1 will be a discussion in Sect. 5 of two Gerˇsgorin type theorems proved by Zhang in [23] for the eigenvalues of a quaternionic matrix. We compute many examples taken from the references, in order to compare our computations with the original on
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