ABSTRACT
This chapter describes the process of transforming from one set of basis vectors to a new set of basis vectors. With multi-dimensional vectors there is no geometrical interpretation for multiplication by a matrix with orthonormal rows. However, one shall refer this operation as a rotation of the coordinate system. The discrete Fourier transform operation can be viewed as a rotation of the basis vector set from the canonical basis set, dn, to a basis set, wk, which consists of sinusoidal waves. The effect of a rotation on a matrix is somewhat more complicated than the effect on a vector. However, understanding this change will provide new insight into a matrix as a system. The chapter examines the ways to rotate the basis vectors to a representation in terms of eigenvectors. Transformation to eigenvector basis, changes the complicated matrix multiplication operation into a simple operation, multiplication by a diagonal matrix.
