In this work, an algebraic method to prove the existence of left eigenvalues for the quaternionic matrix is investigated. The left eigenvalues of a $n\times n$ quaternionic matrix can be derived by solving the zeros of a general quaternionic polynomial of degree $2n$. Using the Study’s determinant, it can be found by solving the zeros of quaternionic polynomials of degree at most $n$ or of rational functions.

中文翻译：

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四元数矩阵左特征值的存在

在这项工作中，研究了一种证明四元数矩阵存在左特征值的代数方法。a的左特征值$n\times n$可以通过求解一般四元数多项式的零点来导出四元数矩阵$2n$. 使用 Study 的行列式，最多可以通过求解次数的四元数多项式的零点来找到$n$或有理函数。