论文标题
八元二次方程
Octonionic Quadratic Equations
论文作者
论文摘要
$ \ mathbb {r} $有四个分区代数，即实数，复数，四元和八元。缺乏换向性和关联性使得很难研究八月的代数和几何特性。例如，询问方程$ x^2+1 = 0 $是否可以解决，而无需指定我们希望解决方案撒谎的字段是没有意义的。等式$ x^2+1 = 0 $在$ \ mathbb {r} $中没有解决方案，也就是说，没有满足此方程的实数。另一方面，所有复数的数字确实满足了该方程的$ \ Mathbb {C} $。如果我们将相同的想法扩展到其他两个规范的代数四季度和八元。唇线黄和Wasin因此得出了计算Quaternionic二次方程的根的显式公式。我们将他们的工作扩展到八氧化节案例，并求解$ x^2+bx+bx+c = 0 $的左八二次方程，其中$ a，b $通常是八元。最后，我们代表$ 2 \ times2 $八元代替矩阵的左光谱作为相应的八二二次二次方程的一组解决方案，该方程的应用是衍生出用于计算八元离子二次方程的根的显式公式的应用。
There are four division algebras over $\mathbb{R}$, namely real numbers, complex numbers, quaternions, and octonions. Lack of commutativity and associativity make it difficult to investigate algebraic and geometric properties of octonions. It does not make sense to ask, for example, whether the equation $x^2+1=0$ is solvable, without specifying the field in which we want the solutions to be lie. The equation $x^2+1=0$ has no solutions in $\mathbb{R}$, which is to say, there are no real numbers satisfying this equation. On the other hand, there are complex numbers which do satisfy this equation in the field $\mathbb{C}$ of all complex numbers. How about if we extend the same idea to other two normed division algebras quaternions and octonions. Liping Huang and Wasin So derive explicit formulas for computing the roots of quaternionic quadratic equations. We extend their work to octonionic case and solve monic left octonionic quadratic equation of the form $x^2+bx+c=0$, where $a,b$ are octonions in general.[ We called this form of quadratic equation as left octonion quadratic equation because we can consider $x^2+xb+c=0$ as a different case due to non-commutativity of octonions]. Finally, we represent the left spectrum of $2\times2$ octonionic matrix as a set of solutions to a corresponding octonionic quadratic equation, which is an application of deriving explicit formulas for computing the roots of octonionic quadratic equations.
