When we have a square matrix of size
n,
A, and we multiply
it by a vector x
from ℂn
to form the matrix-vector product (Definition MVP), the result is another vector
in ℂn.
So we can adopt a functional view of this computation — the act of
multiplying by a square matrix is a function that converts one vector
(x) into
another one (Ax)
of the same size. For some vectors, this seemingly complicated computation is really
no more complicated than scalar multiplication. The vectors vary according to the
choice of A,
so the question is to determine, for an individual choice of
A, if
there are any such vectors, and if so, which ones. It happens in a variety of
situations that these vectors (and the scalars that go along with them) are of
special interest.
We will be solving polynomial equations in this chapter, which raises the
specter of roots that are complex numbers. This distinct possibility is our main
reason for entertaining the complex numbers throughout the course. You might be
moved to revisit Section CNO and Section O.