Question

A question regarding complex eigenvalues. Can someone please explain to me how AX = (1+2i)X is rewritten to the formulae shown on this image: http://i.imgur.com/vhJAN.png

Answer 1

Let & = 1+2i, just so it's notationally neater. Then,

Ax = &x
Ax - &x = 0
(A - &I)x = 0

x = 0 would give us the trivial solution. We're not interested in that. So, A - &I = 0. From there, you just subtract the matrices. Then, take the determinant and you have it in the form you wanted.