Discrete Wavelet Transformations, 2nd Edition

CHAPTER 2VECTORS AND MATRICES

In this chapter we cover some basic concepts from linear algebra necessary to understand the ideas introduced later in the book. Signals and digital audio are typically represented as vectors and digital images are usually described with matrices. Therefore, it is imperative that before proceeding, you have a solid understanding of some elementary concepts associated with vectors and matrices.

If you have completed a sophomore linear algebra class, the first two sections of this chapter will undoubtedly be a review for you. Make sure though, that you take time to understand concepts such as orthogonal vectors and orthogonal matrices. We also discuss matrix multiplication as a way of “processing” another matrix or a vector. Although you may have mastered multiplication with matrices, it is worthwhile to think about the concepts related to matrix multiplication presented in Section 2.2.

The third section of the chapter deals with block (partitioned) matrix algebra. This material is typically not covered in detail in a sophomore linear algebra class but is very useful for analyzing and designing the wavelet transformations that appear in later chapters.

We conclude the chapter with a discussion of convolution and filters. The act of convolving two sequences is at the very heart of signal/image processing and typically one of the convolved sequences possesses special characteristics that influence the convolution product. We will think of this fixed sequence ...