I have problem with calculating left eigenvalues for quaternion Matrices. Let's take a look at article 'Geršgorin type theorems for quaternionic matrices - Fuzhen Zhang':
The right eigenvalues are easy to calculate but im trying to work out how to calculate the left eigenvalues. I dont really know how to compute the det[$\chi_{(A-tI_{n})}]$ - I mean the matrix before calculating determinant
So basically I want to see how calculate $Ax=\lambda x $ and my $\lambda$ will be the left eigenvalue of matrix A. I tried to use complex representation of quaternions for that but I get lost in calculations.
In next article 'On left eigenvalues of a quaternionic matrix Liping Huang a ,∗,1, Wasin Sob,2 ', we have definitions and theorem to calculate the left eigenvalues for 2x2 matrices:
And what I want to know, how to calculate in a raw form left and right eigenvalues and also by using that theorem for 2x2 matrix. I hope someone can explain it to me by calculating them step by step or maybe just some hints. After that maybe i will be able to check more examples if my calculations are correct, that would be awesome.