Quaternionic left eigenvalue problem (Q2569301)

A problem of existence and computing the left eigenvalues of an \(n\times n\) quaternionic matrix is considered. In addition to the known results related to the case \(n=2\) [cf. \textit{L. Huang} and \textit{W. So}, Linear Algebra Appl. 323, No.~1--3, 105--116 (2001; Zbl 0976.15014)], the problem under consideration is solved for the case \(n=3\). It is explicitly shown that all left eigenvalues of a \(3\times 3\) quaternionic matrix can be found by solving quaternionic polynomials of degree of most 3. The known algorithm of \textit{R. Serôdio, E. Pereira}, and \textit{J. Vitória}, [Comput. Math. Appl. 42, No.~8--9, 1229--1237 (2001; Zbl 1050.30037)] for solving a ``standard'' quaternionic polynomial is used. In conclusion, it is pointed out that ``there is no known explicit formula nor algorithm for computing solutions of `general' cubic quaternionic polynomial''.