We have studied the algebraic structure of matrices, and we have studied the way that these algebraic properties are reflected in a geometric model where a matrix represents a transformation from one Euclidean space to another.

In this chapter we shall study how to do this in a different way that cares
more about the geometry adapted to the specific matrix. First we shall take
up square matrices, where we will study *eigenvalues* and
*eigenvectors*. We shall see that some matrices can be reimagined
as if they are diagonal matrices. This idea of *diagonalization*
is powerful, but it doesn't always work. We will study one situation where
we know it will always work in the *spectral theorem* for symmetric
matrices.

Then we shall apply all that we have learned this term and put it together
to study the *singular value decomposition* of a general rectangular
matrix. This will be a good geometric understanding of how a matrix behaves
as a function.