Mathematics
Matrix Calculations
Matrix calculations involve performing various operations on matrices, such as addition, subtraction, multiplication, and finding inverses. Matrices are rectangular arrays of numbers or symbols, and these calculations are fundamental in many areas of mathematics, including linear algebra and calculus. They are also widely used in fields like computer graphics, physics, and engineering for solving systems of equations and representing transformations.
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9 Key excerpts on "Matrix Calculations"
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.
eBook - PDF- Ken Binmore, Joan Davies(Authors)
- 2002(Publication Date)
- Cambridge University Press(Publisher)
1.1 Matrices ♦ A matrix is a rectangular array of numbers – a notation which enables calculations to be carried out in a systematic manner. We enclose the array in brackets as in the examples below: A = ⎛ ⎝ 4 1 0 − 1 3 2 ⎞ ⎠ B = 1 0 − 1 2 1 0 A matrix with m rows and n columns is called an m × n matrix. Thus A is a 3 × 2 matrix and B is a 2 × 3 matrix. 1 Chapter 1. Matrices and vectors A general m × n matrix may be expressed as C = ⎛ ⎜ ⎜ ⎜ ⎝ c 11 c 12 . . . c 1 n c 21 c 22 . . . c 2 n . . . . . . . . . . . . c m 1 c m 2 . . . c mn ⎞ ⎟ ⎟ ⎟ ⎠ where the first number in each subscript is the row and the second number in the subscript is the column. For example, c 21 is the entry in the second row and the first column of the matrix C . Similarly, the entry in the third row and the first column of the preceding matrix A can be denoted by a 31 : column 1 ↓ row 3 −→ ⎛ ⎝ 4 1 0 − 1 3 2 ⎞ ⎠ a 31 = 3 We call the entries of a matrix scalars . Sometimes it is useful to allow the scalars to be complex numbers † but our scalars will always be real numbers . We denote the † See Chapter 13 . set of real numbers by R . Scalar multiplication One can do a certain amount of algebra with matrices and under this and the next few headings we shall describe the mechanics of some of the operations which are possible. The first operation we shall consider is called scalar multiplication . If A is an m × n matrix and c is a scalar, then cA is the m × n matrix obtained by multiplying each entry of A by c . For example, 2 A = 2 ⎛ ⎝ 4 1 0 − 1 3 2 ⎞ ⎠ = ⎛ ⎝ 2 × 4 2 × 1 2 × 0 2 × − 1 2 × 3 2 × 2 ⎞ ⎠ = ⎛ ⎝ 8 2 0 − 2 6 4 ⎞ ⎠ Similarly, 5 B = 5 1 0 − 1 2 1 0 = 5 0 − 5 10 5 0 Matrix addition and subtraction If C and D are two m × n matrices, then C + D is the m × n matrix obtained by adding corresponding entries of C and D . Similarly, C − D is the m × n matrix obtained by subtracting corresponding entries.
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.
- 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- 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- John L. Teall(Author)
- 1999(Publication Date)
- Praeger(Publisher)
4 Matrix Mathematics 4.A: MATRICES, VECTORS AND SCALARS A matrix is defined as an ordered rectangular array of numbers. A matrix enables one to represent a series of numbers as a single object, thereby providing for convenient systematic methods for completing repetitive computations. The following are examples of matrices: The dimensions of a matrix are given by the ordered pair m x n, where m is the number of rows and n is the number of columns in the matrix. Thus, A is 3 x 2, B is 2 x 2, c is 2 x 1, and d is 1 x 1. Each number in a matrix is referred to as an element. The symbol a^ denotes the element in Row i and Column j of Matrix A, b l} denotes the element in Row i and Column j of Matrix B, and so on. Thus, a 32 is 4 and c 2l = 3. There are specific terms denoting special types of matrices. For example, a vector is a matrix with either only one row or one column. Thus, the dimensions of a vector are 1 x n or m x 1. Matrix c above is a column vector; a 1 x n matrix is a row vector. A scalar is a matrix with exactly one element. Matrix d is a scalar. A square matrix has the same number of rows and columns (m = n). Matrices B and d are square matrices. A symmetric matrix is a square matrix where c i(j equals c jti for all i and j ; that is, the i'th element in each row equals the j'th element in each column. Scalar d and matrices H, I, and J below are all symmetric matrices. A diagonal matrix is a symmetric matrix whose elements off the principal diagonal are zero, where the principle diagonal contains the series of elements where i = j . Scalar d and Matrices H, and I below are all diagonal matrices. An identity or unit matrix is a diagonal matrix 50 Chapter 4 consisting of ones along the principal diagonal.
- 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).
- Ron Larson(Author)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
Matrices Clockwise from top left, Cousin_Avi/Shutterstock.com; Goncharuk/Shutterstock.com; Gunnar Pippel/Shutterstock.com; Andresr/Shutterstock.com; nostal6ie/Shutterstock.com 2.1 Operations with Matrices 2.2 Properties of Matrix Operations 2.3 The Inverse of a Matrix 2.4 Elementary Matrices 2.5 Markov Chains 2.6 More Applications of Matrix Operations Information Retrieval (p. 58) Flight Crew Scheduling (p. 47) Beam Deflection (p. 64) Computational Fluid Dynamics (p. 79) Data Encryption (p. 94) 2 39 Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 40 Chapter 2 Matrices 2.1 Operations with Matrices Determine whether two matrices are equal. Add and subtract matrices and multiply a matrix by a scalar. Multiply two matrices. Use matrices to solve a system of linear equations. Partition a matrix and write a linear combination of column vectors. EQUALITY OF MATRICES In Section 1.2, you used matrices to solve systems of linear equations. This chapter introduces some fundamentals of matrix theory and further applications of matrices. It is standard mathematical convention to represent matrices in any one of the three ways listed below. 1. An uppercase letter such as A , B , or C 2. A representative element enclosed in brackets, such as [ a ij ] , [ b ij ] , or [ c ij ] 3. A rectangular array of numbers bracketleft.alt2 a 11 a 21 uni22EE.alt2 a m 1 a 12 a 22 uni22EE.alt2 a m 2 . . . . . . . . . a 1 n a 2 n uni22EE.alt2 a mn bracketright.alt2 As mentioned in Chapter 1, the matrices in this text are primarily real matrices . That is, their entries are real numbers. Two matrices are equal when their corresponding entries are equal. Definition of Equality of Matrices Two matrices A = [ a ij ] and B = [ b ij ] are equal when they have the same size ( m × n ) and a ij = b ij for 1 ≤ i ≤ m and 1 ≤ j ≤ n .
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