Discrete Wavelet Transformations: An Elementary Approach with Applications

CHAPTER 2

VECTORS AND MATRICES

In this chapter we cover some basic concepts from linear algebra necessary to understand the ideas covered later in the book. Signals and digital audio are typically represented as vectors, and digital images are usually described using 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 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 the matrix as a way of “processing” another matrix or a vector via matrix multiplication. 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 final section of the chapter deals with block (partitioned) matrix arithmetic. 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.

2.1   VECTORS, INNER PRODUCTS, AND NORMS

We remember vectors from a linear algebra course. In ², vectors take the form

where υ₁ and υ₂ are any real numbers. You may recall the idea of vectors in ⁿ: