Mathematics
Matrices
Matrices are arrays of numbers or symbols arranged in rows and columns. They are used to represent and solve systems of linear equations, transformations, and other mathematical operations. Matrices are fundamental in various fields such as physics, computer graphics, and engineering, and they provide a concise way to organize and manipulate data in mathematical calculations.
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12 Key excerpts on "Matrices"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
Matrices find many applications. Physics makes use of Matrices in various domains, for example in geometrical optics and matrix mechanics; the latter led to studying in more detail Matrices with an infinite number of rows and columns. Graph theory uses Matrices to keep track of distances between pairs of vertices in a graph. Computer graphics uses Matrices to project 3-dimensional space onto a 2-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives of functions or exponentials to Matrices. The latter is a recurring need in solving ordinary differential equations. Serialism and dodecaphonism are musical movements of the 20th century that use a square mathematical matrix to determine the pattern of music intervals. A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old but still an active area of research. Matrix decomposition methods simplify computations, both theoretically and practically. For sparse Matrices, specifically tailored algorithms can provide speedups; such Matrices arise in the finite element method, for example. Definition A matrix is a rectangular arrangement of numbers. For example, An alternative notation uses large parentheses instead of box brackets: The horizontal and vertical lines in a matrix are called rows and columns , respectively. The numbers in the matrix are called its entries or its elements . To specify the size of a matrix, a matrix with m rows and n columns is called an m -by-n matrix or m × n matrix, while m and n are called its dimensions . The above is a 4-by-3 matrix. ________________________ WORLD TECHNOLOGIES ________________________ A matrix with one row (a 1 × n matrix) is called a row vector, and a matrix with one column (an m × 1 matrix) is called a column vector.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
Matrices find many applications. Physics makes use of Matrices in various domains, for example in geometrical optics and matrix mechanics; the latter led to studying in more detail Matrices with an infinite number of rows and columns. Graph theory uses Matrices to keep track of distances between pairs of vertices in a graph. Computer graphics uses Matrices to project 3-dimensional space onto a 2-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives of functions or exponentials to Matrices. The latter is a recurring need in solving ordinary differential equations. Seria-lism and dodecaphonism are musical movements of the 20th century that use a square mathematical matrix to determine the pattern of music intervals. A major branch of numerical analysis is devoted to the development of efficient algori-thms for matrix computations, a subject that is centuries old but still an active area of research. Matrix decomposition methods simplify computations, both theoretically and practically. For sparse Matrices, specifically tailored algorithms can provide speedups; such Matrices arise in the finite element method, for example. Definition A matrix is a rectangular arrangement of numbers. For example, An alternative notation uses large parentheses instead of box brackets: The horizontal and vertical lines in a matrix are called rows and columns , respectively. The numbers in the matrix are called its entries or its elements . To specify the size of a matrix, a matrix with m rows and n columns is called an m -by-n matrix or m × n matrix, while m and n are called its dimensions . The above is a 4-by-3 matrix. ________________________ WORLD TECHNOLOGIES ________________________ A matrix with one row (a 1 × n matrix) is called a row vector, and a matrix with one column (an m × 1 matrix) is called a column vector.
- David C. Vella(Author)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
2 Matrix Algebra In Chapter 1, we encountered Matrices as a device to streamline the process of elimination to solve systems of linear equations. However, Matrices have a life of their own beyond being a bookkeeping aid for elimination. In this chapter, the algebra of Matrices and some simple applications are introduced. A large part of the rest of the book illustrates more applications of Matrices. They are quite useful, and their theory is rich and deep. This book is a mere introduction, and it is not very theoretical; if the reader is interested in learning more about Matrices, the next step would be to take a course in linear algebra, where the theory is developed more carefully and completely than it is here. 2.1 Matrices A matrix (plural, Matrices) is a rectangular array of numbers, called entries or elements. As we’ll see later in this text, it is sometimes useful to allow the entries of a matrix to be ordered pairs instead of individual numbers. But for now, we’ll stick to the given definition. The size or dimensions of a matrix are the number of (horizontal) rows and (vertical) columns (in that order), so that a 2 × 3 matrix is one with two rows and three columns. Capital letters will be used to stand for a matrix, and the entries inside are referred to by the corresponding lowercase letters. Different entries are distinguished by their address, which is a double subscript indicating in which row and column it appears. Thus, a ij refers to the element of the matrix A in the i th row and j th column. For example, in the matrix A = 2 1 3 0 −1 3 , we have a 11 = 2, a 13 = 3 = a 23 , etc. The general pattern (for a 3 × 4 matrix, for example) is A = A 3×4 = ⎡ ⎣ a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 ⎤ ⎦ . (Notice that we sometimes write the dimensions of the entire matrix A as a subscript on A.) If there are more than nine rows or columns in A, we use commas to separate the row and column subscripts.
eBook - PDF- David W Lewis(Author)
- 1991(Publication Date)
- WSPC(Publisher)
1 Chapter 1 Matrices AND LINEAR EQUATIONS A familiarity with Matrices is necessary nowadays in many areas of mathematics and in a wide variety of other disciplines. Areas of mathematics where Matrices occur include algebra, differential equations, calculus of several variables, probability and statistics, optimization, and graph theory. Other disciplines using matrix theory include engineering, physical sciences, biological sciences, economics and management science. In this first chapter we give the fundamentals of matrix algebra, determinants, and systems of linear equations. At the end of the chapter we give some examples of situations in mathematics and other disciplines where Matrices arise. 1.1 Matrices and matrix algebra A matrix is a rectangular array of symbols. In this book the symbols will usually be either real or complex numbers. The separate elements of the array are known as the entries of the matrix. Let m and n be positive integers. An rrun matrix A consists of m rows and n columns of numbers written in the following manner. f a ,, a n . ■ . a, ) 11 12 In a a . . . a 21 22 2n A = * a a . . . a I ml m2 mnl 2 We often write A = (a..) for short. The entry a lies in the i-th row and ij ij the j-th column of the matrix A. Two mxn Matrices A = (a.) and B = (b.) are equal if and only if all ij u the corresponding entries of A and B are equal. i.e. a.. = b.. for each i and j . ij ij The sum of the mxn Matrices A = (a.) and B = (b.) is the mxn matrix denoted A + B which has entry a.. + b.. in the (ij)-place for each i,j. Let X be a scalar (i.e. a real or complex number) and let A = (a.) be an mxn matrix. The scalar multiple of A by X is the nun matrix denoted XA which has entry Xajn the (ij)-place for each i,j. 1.1.2 Proposition The following properties hold. (i) A + B = B + A for all mxn Matrices A and B, i.e. addition of Matrices is commutative. (ii) (A + B) + C = A + (B + C) for all nun Matrices A,B, and C, i.e.addition of Matrices is associative.
eBook - PDF- Marvin Tobias(Author)
- 2022(Publication Date)
- Springer(Publisher)
1 C H A P T E R 1 Matrix Fundamentals 1.1 DEFINITION OF A MATRIX A matrix is defined to be a rectangular array of functional or numeric elements, arranged in row or column order. Most important in this definition is that (at most) two subscripts, or indices, are required to identify a given element: a row subscript, and a column subscript. That is, a matrix is a 2-dimensional array. Included within the definition are arrays in which the maximum value of one, or both subscripts is unity. For example, a single “list” of elements, arranged in a single row or column, is referred to as a “row” or “column” matrix. Even a single element may be referred to as a one-by-one (i.e., 1X1) matrix. By way of illustration, the following matrix, “A,” is diagrammed: A = ⎡ ⎢ ⎢ ⎢ ⎣ a 11 a 12 a 13 · · · a 1n a 21 a 22 · · · · · · a 2n · · · · · · · · · . . . · · · a m1 a m2 a m3 · · · a mn ⎤ ⎥ ⎥ ⎥ ⎦ The above rectangular matrix has m rows, and n columns. The purpose of this book will be to discuss and define the arithmetic (and mathematics) of such arrays. Practical applications will be discussed, in which the array will often be viewed and manipulated as a single entity. Once the notation of Matrices is learned, there follows a very large advantage in being able to work with the array as an entity, without being encumbered with the arithmetic manipulation of the numeric values inside. That is, one of the big advantages is that of “bookkeeping.” Carrying this illustration further, we write an m-by-n set of linear algebraic equations as: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a 11 x 1 + a 12 x 2 + a 13 x 3 + · · · + a 1n x n = c 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + · · · + a 2n x n = c 2 · · · · · · · · · · · · · · · = · · · a m1 x 1 + a m2 x 2 + a m3 x 3 + · · · + a mn x n = c m . (1.1) The above defines a set of m-equations in n-unknowns, the solutions to which will be explored in a later chapter. Right now, the point is to compare the equation set (1.1) with the definition of the m row by n column matrix above.
eBook - PDF- Magar Mager(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
CHAPTER 2 Matrices 1. Introduction This chapter is the first of five designed to build the required math-ematical statistical background to handle the equations encountered in this type of work and to provide a framework within which we may systematically analyze data. From the introductory chapter we see that we are faced with experimental systems of linear equations, which could be written in the notation of Matrices thus facilitating the manipulation of the equations and providing a compact notation. 2. Matrices A matrix is a rectangular array of numbers enclosed by a pair of brackets and is depicted in its most general form as I « 2 1 « 1 2 « 2 2 *23 (2-1) 20 3. Rules of Matrix Algebra 21 This array has m rows and η columns and denotes the element in the ith row and the jth column. T h u s we note that all elements in the second row have 2 as first subscript and all elements in the third column have 3 as second subscript. Similarly for other rows and columns. Sometimes Matrices are written for simplicity as («y). T h e matrix A denoted by (2-1) is called an w X m matrix. For example Γ5 -2 r 3 6 -4 (2-2) 3 -7 1 11 -2 8. (2-3) the matrix (2-2) is a 3 X 2 matrix and the elements a 13 = 1 and a 22 = 6. T h e matrix (2-3) is a 3 X 3 matrix and the elements a S2 = — 2, a 12 = 3, and so on. When η = 1 for all elements in a given matrix the matrix reduces to a single column and we refer to it as a column vector. This is illustrated by « 1 1 « 1 2 (2-4) Similarly, if m = 1, the matrix reduces to a single row which we refer to as a row vector. This is illustrated by [ « 1 1 > « 1 2 > « 1 3 > · · · > « l n ] (2-5) 3. Rules of Matrix Algebra 3.1 EQUALITY OF Matrices Two Matrices are equal if they have the same number of elements and if their corresponding elements are equal. T h u s given a typical element of the matrix A and the corresponding element of a matrix Β then the Matrices are equal if and only if « i j ; = for f = l , . . . , n , j =!,...> m (2-6)
- Shawna Lockhart, Eric Tilleson(Authors)
- 2019(Publication Date)
- SDC Publications(Publisher)
INTRODUCTION 83 5 Objectives When you have com- pleted this tutorial, you will be able to 1. Explain the difference between an array, a matrix, a vector, and a scalar. 2. Specify an array in MATLAB using brackets and semi- colons. 3. Understand the concepts of a diagonal, identity, and magic matrix. 4. Add, subtract, multiply, and divide a matrix by a scalar or vector. 5. Raise the values in a matrix to a scalar power or a vector of powers. 6. Use MATLAB functions to create test Matrices. 7. Seed a sequence of random numbers so that the sequence can be replicated. 8. Recognize when two Matrices meet the requirements for matrix multiplication. 9. Perform a matrix multiplication manually. 10. Have a rudimentary understanding of an inverse matrix and a matrix determinant. Matrices Introduction You can use MATLAB to write a program that is as complex and useful as any in another programming language, as well as easily visualize the data and create 3D graphs, but MATLAB’s ability to deal with Matrices sets it apart. The branch of mathematics called linear algebra is built on the interac- tion and manipulation of Matrices to solve complex problems. Linear algebra is generally taught following the calculus series. Linear algebra and matrix mathematics are used in engineering (modeling circuits, robotic motion), physics (quantum mechanics), computer graphics (reflections, 3D projections to 2D), and digital photography (device- based color correction), just to scratch the surface. We can’t teach linear algebra here, but you’ll get a glimpse of some of its potential uses and gain an understanding of Matrices. When you have a need for solutions using Matrices, you’ll find MATLAB invaluable. What Is a Matrix? As you learned previously, a matrix (plural: Matrices) is a two-dimen- sional, rectangular array consisting of rows and columns. A vector is a matrix with only one row (1 x N) or one column (N x 1).
- Sudipto Banerjee, Anindya Roy(Authors)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
Alternatively, one could create one augmented array 4 7 2 0 2 -6 -10 0 1 1 4 6 4 5 0 1 2 Matrices, VECTORS AND THEIR OPERATIONS with a “ | ” to distinguish the right-hand side of the linear system. We will return to solving linear equations using Matrices in Chapter 2 . More gener-ally, rectangular arrays are often used as data structures to store information in com-puters. If we can define algebraic operations on such arrays in a meaningful way, then we can not only use arrays as mere storage devices but also for solving linear systems of equations in computers. Rectangular arrays of numbers are called Matrices . When the array is a single row or column it is called a vector . In this chapter we introduce notations and develop some algebraic operations involving Matrices and vectors that will be used extensively in subsequent chapters. 1.1 Basic definitions and notations Definition 1.1 A matrix of order (or dimension) m × n is a collection of mn items arranged in a rectangular array with m rows and n columns as below: A = a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a m 1 a m 2 . . . a mn or a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a m 1 a m 2 . . . a mn . The individual items in a matrix are called its elements or entries . The elements of a matrix are often called scalars . In this book, unless explicitly stated otherwise, the elements of a matrix will be assumed to be real numbers, i.e., a ij ∈ < for i = 1 , . . . , m and j = 1 , . . . , n . A matrix is often written in short form as A = { a ij } m,n i,j =1 , or simply as { a ij } when the dimensions are obvious. We also write [ A ] ij or ( A ) ij to denote the ( i, j ) -th element of A . When the order of a matrix needs to be highlighted, we will often write A m × n to signify an m × n matrix. Example 1.2 Suppose we have collected data on seven rats, each of whom had its heights measured weekly for five weeks.
No longer available |Learn more- Xuan LIU, Zhi ZHAO, Wei-Hui LIU, Xiao-Qing JIN(Authors)
- 2022(Publication Date)
- EDP Sciences(Publisher)
Chapter 1 Linear Systems and Matrices “No beginner’s course in mathematics can do without linear algebra.” — Lars Gårding “Matrices act. They don’t just sit there.” — Gilbert Strang Solving linear systems (a system of linear equations) is the most important problem of linear algebra and possibly of applied mathematics as well. Usually, information in a linear system is often arranged into a rectangular array, called a “matrix”. The matrix is particularly important in developing computer programs to solve linear systems with huge sizes because computers are suitable to manage numerical data in arrays. Moreover, Matrices are not only a simple tool for solving linear systems but also mathematical objects in their own right. In fact, matrix theory has a variety of applications in science, engineering, and mathematics. Therefore, we begin our study on linear systems and Matrices in the first chapter. 1.1 Introduction to Linear Systems and Matrices Let denote the set of real numbers. We now introduce linear equations, linear systems, and Matrices. 1.1.1 Linear equations and linear systems We consider a 1 x 1 + a 2 x 2 + · · · + a n x n = b, where a i ∈ (i = 1, 2, . . . , n) are coecients, x i (i = 1, 2, . . . , n) are variables (unknowns), n is a positive integer, and b ∈ is a constant. An equation of this form is called a linear equation, in which all variables occur to the first power. When b = 0, the linear equation is called a homogeneous linear equation. A 2 Chapter 1 Linear Systems and Matrices sequence of numbers s 1 , s 2 , . . . , s n is called a solution of the equation if x 1 = s 1 , x 2 = s 2 , . . . , x n = s n such that a 1 s 1 + a 2 s 2 + · · · + a n s n = b. The set of all solutions of the equation is called the solution set of the equation. In the book, we always use example(s) to make our points clear. Example We consider the following linear equations: (a) x + y = 1.
eBook - PDF- Marcel Maeder, Yorck-Michael Neuhold(Authors)
- 2007(Publication Date)
- Elsevier Science(Publisher)
It is helpful to distinguish Matrices, vectors, scalars and indices by typographic conventions. Matrices are denoted in boldface capital characters ( A ), vectors in boldface lowercase ( a ) and scalars in lowercase italic characters ( s ). For indices, lower case characters are used (i). The symbol 't' indicates matrix and vector transposition ( A t , a t ). All chemical applications discussed later in this book will deal exclusively with real numbers. Thus, we introduce matrix algebra for real numbers only and do not include Matrices formed by complex numbers. 2.1 Matrices, Vectors, Scalars A matrix is a rectangular array of numbers. e.g. ª a 1,1 a 1, j a 1, n º « » # % # « » A « a i ,1 a , a j n » (2.1) i j , « » # % # « » « a a m j a , » ¬ m ,1 , m n ¼ The size of a matrix is defined by its number of rows, m , and number of columns, n . We refer to the dimensions by m u n . In Matlab, the appropriate notation is [m,n]=size(A) . For any matrix A , a i , j is the element in row i ( i =1… m ) and column j ( j =1… n ). Vectors and scalars can be seen as special cases of Matrices where one or both dimensions have collapsed to 1. Thus, a row vector represents a 1 u n matrix, a column vector an m u 1 matrix and a scalar a 1 u 1 matrix. Matlab is based on the philosophy that Everything is a Matrix . In order to visualise Matrices, column and row vectors, it is convenient to use rectangles, vertical and horizontal lines, as outlined in Figure 2-1. Figure 2-1. A Matrix, a column vector and a row vector Sometimes it is helpful to specifically distinguish between row and column vectors. In such instances, we borrow Matlab's colon (:) notation. A vector x 9 Matrix Algebra is represented by x 1,: if it is a row vector (the row dimension is 1) and by x :,1 if it is a column vector (the column dimension is 1).
- Will H. Moore, David A. Siegel(Authors)
- 2013(Publication Date)
- Princeton University Press(Publisher)
Part IV Linear Algebra Chapter Twelve Fun with Vectors and Matrices You first encountered mathematics as whole numbers and learned how to add, subtract, multiply and divide them. That field of inquiry is arithmetic. A loose distinction between arithmetic and algebra is that the latter replaces numbers with variables. So, for example, we can let the variable x take any value within the set of whole numbers. What you may not know is that the algebra you learned in middle school (in the United States) could actually be called scalar algebra, which is to say algebra concerning single variables. Scalar algebra in this usage is a subset of vector algebra, which is itself a subset of matrix algebra. Loosely speaking, vector algebra deals with one-dimensional collections of vari- ables, while matrix algebra deals with two-dimensional collections of variables. Now, instead of a single scalar x, we might have the vector x, which contains the set of scalars x 1 , x 2 , . . . , x i , . . . , x n , with each x i a different element of the vector x. Or we might have the matrix X, which contains the set of vectors x 1 , . . . , x m or, equivalently, the set of scalars x 11 , . . . , x 1m , x 21 , . . . , x 2m , . . . , x n1 , . . . , x nm . A scalar variable thus has one element, a vector n elements, and a matrix n × m elements. One useful way to think about the three sets of algebras is that they are nothing more than different means of representing variables that we want to manipulate. A matrix holds more information than a vector, which holds more information than a scalar. It is often quite sufficient to represent variables in scalar form, and when that is all we need to conduct our business, we turn to scalar algebra. At other times, however, it turns out that there is an advantage to writing down a problem using vectors rather than limiting ourselves to a scalar representation of variables, and in such circumstances we need to understand vector algebra.
eBook - PDF- Al Cuoco, Kevin Waterman, Bowen Kerins, Elena Kaczorowski(Authors)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
Chapter 5 Matrices as Functions 13. Suppose that A and B are linearly independent vectors in R 2 and that C = xA + yB for numbers x and y . C = xA + yB yB B A xA Show that ←− Here, det( A, B ) means the determinant of the matrix whose columns are A and B . x = det( C,B ) det( A,B ) and y = det( A,C ) det( A,B ) a. using algebra b. using geometry 14. In Exercise 18 from Lesson 2.5, you saw that if A,B , and C are vectors in R 3 , no two of which are collinear, then the volume of the parallelepiped determined by A,B, and C is | C · ( A × B ) | . B O h A C A × B It would make sense, then, to extend the definition of determinant ←− You’ll treat general deter-minants in depth in Chap-ter 9. to 3 × 3 Matrices by defining the determinant of a 3 × 3 matrix whose columns are A , B , and C to be C · ( A × B ). Find the determinant of each matrix: a. det ⎛ ⎝ 1 2 3 4 5 6 7 8 10 ⎞ ⎠ b. det ⎛ ⎝ 2 2 3 8 5 6 14 8 10 ⎞ ⎠ c. det ⎛ ⎝ 2 1 3 5 4 6 8 7 10 ⎞ ⎠ d. det ⎛ ⎝ 6 2 3 7 5 6 8 8 10 ⎞ ⎠ e. det ⎛ ⎝ 1 + 6 2 3 4 + 7 5 6 7 + 8 8 10 ⎞ ⎠ f. det ⎛ ⎝ 1 2 3 4 5 6 7 8 9 ⎞ ⎠ 15. a. Which of the properties in Theorems 5.7 and 5.8 extend to 3 × 3 Matrices? b. Take It Further. Proofs? 246 5.5 Image, Pullback, and Kernel 5.5 Image, Pullback, and Kernel Most of the matrix functions you have been working with thus far have been represented by square Matrices. However, nonsquare Matrices can also act as functions. With nonsquare Matrices, the outputs are in a different dimension from the inputs, so lines and shapes acted upon by those Matrices may also change dimensions. In this lesson, you will learn how to • find the image of a linear map using matrix multiplication • find and interpret geometrically the set of pullbacks of a vector under a given matrix • understand the relationship between the kernel of a matrix and the set of pullbacks of a vector under that matrix For You to Do 1.
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