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.