# Left (right) eigenvalues of quaternion Matrix

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':

Click

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

Example

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:

theorem

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.

• Obvious question: what does $\chi_A$ mean? I assume it means turning quaternionic matrices (including column vectors) into double-sized complex matrices. If so, you need to decide if you're treating $\Bbb H^n$ as a left or right complex vector space. – arctic tern Mar 13 at 2:24
• I have a problem with calculating left eigenvalues of quaternion matrix without using complex representation. – Michal Mar 13 at 12:46
• You neither confirmed nor denied my suspicion that $\chi_A$ represents turning quaternionic matrices into complex matrices. If my suspicion is right, then why bother linking that image when you're specifically trying to do it a different way (and what other way are you considering)? If my suspicion is wrong, then what does $\chi_A$ represent? – arctic tern Mar 13 at 14:50
• $\chi_{A}$ your assumption was right. But my problem is, first we create B=A-tIn matrix then we transform it into 2n x 2n complex matrix using $B=B_{2}+B_{2}j$ I guess – Michal Mar 14 at 16:18
• I want to calculate eigenvalues like in real case by solving the equation Ax=tx, Ax=tx and left eigenvalues using 'click' image for left eigenvalues. – Michal Mar 14 at 23:12