Complex numbers are generally thought of as a system of arithmetic invented to
solve polynomials, which turned out to also model rotation and dilation in the
plane. Conversely, rotations and dilations in the plane are concrete linear
operations that have the same algebraic properties as these complex numbers,
and, like all operations that can be represented as a square matrix, satisfy a
polynomial called their characteristic polynomial – this characteristic
polynomial being exactly the kind of polynomial complex numbers were devised to
solve. Now in order to do anything with these quadratic characteristic
polynomials we need to solve them using complex numbers, and if complex numbers
are supposed to be rotations and dilations, then really we are using rotations
and dilations in some complex plane, in order to understand linear operations
in some other space. What if we tried to understand the effect of a linear
operation directly in terms of the planes that it acts on, rather than planes
consisting of additional imaginary
axes? In this article I shall
describe a purely geometric interpretation of the quadratic factors in a
matrix's characteristic polynomial, in terms of a system of vector equations
related to, but distinct from, the eigenvector problem.