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Vectors and matrices are useful in representing multivariate data, and they occur naturally in working with linear equations or when expressing linear relationships among objects. Numerical algorithms for a variety of tasks involve matrix and vector arithmetic. An optimization algorithm to find the minimum of a function, for example, may use a vector of first derivatives and a matrix of second derivatives; and a method to solve a differential equation may use a matrix with a few diagonals for computing differences.

In this chapter we discuss a wide range of basic topics related to vectors of real numbers. Some of the properties carry over to vectors over other fields, such as complex numbers, but the reader should not assume this. Occasionally, for emphasis, we will refer to “real” vectors or “real” vector spaces, but unless it is stated otherwise, we are assuming the vectors and vector spaces are real. The topics and the properties of vectors and vector spaces that we emphasize are motivated by applications in the data sciences.

In this chapter, we build on the notation introduced on page 5, and discuss a wide range of basic topics related to matrices with real elements. Some of the properties carry over to matrices with complex elements, but the reader should not assume this. Occasionally, for emphasis, we will refer to “real” matrices, but unless it is stated otherwise, we are assuming the matrices are real.

The operations of differentiation and integration of vectors and matrices are logical extensions of the corresponding operations on scalars.

In most applications of linear algebra, problems are solved by transformations of matrices. A given matrix that represents some transformation of a vector is transformed so as to determine one vector given another vector.

There are two general methods of solving a system of linear equations: direct methods and iterative methods. A direct method uses a fixed number of computations that would in exact arithmetic lead to the solution; an iterative method generates a sequence of approximations to the solution. Iterative methods often work well for very large sparse matrices. We first consider a characteristic of the problem that affects how easy it is to solve the system accurately.

Before we discuss methods for computing eigenvalues, we mention an interesting observation. A given th-degree polynomial () is the characteristic polynomial of some matrix. The companion matrix of equation (3.177) is one such matrix.

In previous chapters, we defined a number of special matrices, such as symmetric matrices, banded matrices, elementary operator matrices, and so on. In this chapter, we will discuss some of these matrices in more detail and also introduce some other special matrices and data structures that are useful in statistics.

Data come in many forms. In the broad view, the term “data” embraces all representations of information or knowledge. There is no single structure that can efficiently contain all of these representations. Some data are in free-form text (for example, the Federalist Papers, which was the subject of a famous statistical analysis), other data are in a hierarchical structure (for example, political units and subunits), and still other data are encodings of methods or algorithms. (This broad view is entirely consistent with the concept of a “stored-program computer”; the program is the data.)