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4_Matrices

# 4_Matrices - Chapter 4 Review of Matrix Theory Chapter...

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53 Chapter 4 Review of Matrix Theory Chapter Outline 4.1 BASIC DEFINITIONS AND ELEMENTARY OPERATIONS . ................... 54 4.1.1 Definitions. ...................................................................................................................................... 4.1.2 Elementary Operations with Matrices. ............................................................................... 56 4.2 VECTORS . ............................................................................................. 60 4.2.1 Definition. ........................................................................................................................................ 4.2.2 Eigenvectors and Eigenvalues. .............................................................................................. 61 4.3 HOMEWORK FOR CHAPTER 4. ............................................................ 64 In this chapter we review some of the elementary definitions and results from matrix theory. The intention is to provide the background for concepts involving state space representations. The required concepts are mainly the elementary operations of matrices, vectors, and eigenvalues. Computer Usage : Many CAD packages for system analysis are based on matrices. This format has shown itself to be both flexible and powerful. This chapter was written to provide some of the basic background needed to use these CAD packages, in particular MATLAB. Hence, the discussion of partitioned matrices, which normally wouldn’t be included with this material. Summary of Sections Section 4.1: We define the basic quantities associated with matrices and we introduce elementary matrix operations. Section 4.2: We introduce vectors. Eigenvalues and eigenvectors are also defined. Coverage of the Text This chapter only requires the notion of a complex number from Chapter 3.

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54 Chapter 4 Review of Matrix Theory 4.1 BASIC DEFINITIONS AND ELEMENTARY OPERATIONS 4.1.1 Definitions We begin by defining an array of numbers. Definition 4.1.1 : An n x m matrix A is the two-dimensional, n x m array of numbers A aa a a a m m nn n m = 11 12 1 21 22 2 12 L L MM O M L . Each entry a ij is called an element of A . We say the dimensions of A are n x m . ▲▲ Notation : The elements a ij of the matrix are located by their row index i and column index j . Sometime for convenience, we denote the matrix by A = [ a ij ]. With this notation it is assumed that dimensions of the matrix, n rows and m columns, are known. Note : The dimensions of the matrix play an important role in matrix theory, and in the implementation of matrix calculations on a computer. Definition 4.1.2 : A square matrix is a matrix where the number of rows equals the number of columns. The dimensions of a square matrix are n x n . We say the matrix has order n . In this text we will be most often concerned with matrices filled with real or complex numbers. However, we need not confine the elements of the matrix to real or complex numbers. The elements of A can be chosen from any set of objects. The elements can also be functions. We will use both matrices whose elements are polynomials and those matrices whose elements are real functions. Terminology : It is commonplace to refer to the matrix by type of elements in the matrix. Matrices with real numbers as elements are called real matrices. Matrices with complex elements are called complex matrices. Matrices with polynomials as entries are called polynomial matrices. And so on.
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4_Matrices - Chapter 4 Review of Matrix Theory Chapter...

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