how is Hamiltons's relation of quaternions related to the charateristic equation of a mattix?

Answer 1

i, j, and k can be represented as vectors in R^3 (notation is varied). You can also represent them as matrices in M(2,R) (the set of 2x2 matrices over R). Any nxn matrix can be thought of as a map from R^n to R^n where the matrix is the transition map.

Are you looking at finding characteristic equations for 3x3 matrices or any nxn matrix?

If it is 3x3, I would use i, j, k as the negatives of the basis of R^3 and then you should be able to write each of the columns of the matrix as a linear combination of these. I am not sure how to get the quaternions related to the characteristic equations.