Physics
Matrices in Physics
Matrices in physics are used to represent physical quantities and transformations in a concise and organized manner. They are essential in quantum mechanics, where they represent observables and operators. Matrices also play a crucial role in solving systems of linear equations and analyzing the behavior of physical systems.
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8 Key excerpts on "Matrices in Physics"
eBook - ePub- Fletcher Dunn, Ian Parberry(Authors)
- 2011(Publication Date)
- A K Peters/CRC Press(Publisher)
Chapter 4Introduction to Matrices
Unfortunately, no one can be told what the matrix is. You have to see it for yourself.— Morpheus in The Matrix (1999)Matrices are of fundamental importance in 3D math, where they are primarily used to describe the relationship between two coordinate spaces. They do this by defining a computation to transform vectors from one coordinate space to another.This chapter introduces the theory and application of matrices. Our discussion will follow the pattern set in Chapter 2 when we introduced vectors: mathematical definitions followed by geometric interpretations.- Section 4.1 discusses some of the basic properties and operations of matrices strictly from a mathematical perspective. (More matrix operations are discussed in Chapter 6 .)
- Section 4.2 explains how to interpret these properties and operations geometrically.
- Section 4.3 puts the use of matrices in this book in context within the larger field of linear algebra.
4.1 Mathematical Definition of Matrix
In linear algebra, a matrix is a rectangular grid of numbers arranged into rows and columns . Recalling our earlier definition of vector as a one-dimensional array of numbers, a matrix may likewise be defined as a two-dimensional array of numbers. (The “two” in “two-dimensional array” comes from the fact that there are rows and columns, and should not be confused with 2D vectors or matrices.) So a vector is an array of scalars, and a matrix is an array of vectors.This section presents matrices from a purely mathematical perspective. It is divided into eight subsections.
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.
eBook - PDF- Peter Luger(Author)
- 2011(Publication Date)
- De Gruyter(Publisher)
1 Theoretical Basis 1.1 Matrices, Vectors 1.1.1 Introduction The first part of this chapter is concerned with some mathematics which will be used in the later chapters of this book. We assume that the fundamentals of arithmetic and of integral and differential calculus are well-known to the reader, but students of chemistry often have difficulties with the theory of vector and matrix algebra. Since we will make frequent use of these mathematical formalisms, the most important properties of vectors and matrices are briefly discussed. 1 . 1 . 2 Matrices, Determinants, Linear Equations A rectangular array, A, arranged in the form is called a matrix. The elements a ik can be arbitrary numbers. If the number of rows is m and the number of columns is n, the matrix is said to be of the order m χ η. If m = η the matrix is called a square matrix of order n. The index i of the element a ik indicates its row and the index k the corresponding column. As will be shown in the next chapter, the matrix formalism is a very convenient way to describe vector operations, vector transformations and it provides a very elegant method for solving linear equations. The introduction of matrices requires a knowledge of matrix algebra. First, we define the basic arithmetic operations of matrices. The equality ofmatrices. Two matrices are said to be of equal type if their numbers of rows and columns are equal. Two matrices are equal, if they are of equal type and if all elements in corresponding rows and columns are equal. A— a 31 a 32 a 33 ... a 3n a l l a 1 2 a 1 3 · · · a l n a 21 a 22 a 23 ... a 2n a m l a m 2 a m 3 · · · a m n / Example: (a) The matrices 2 Theoretical Basis A = 4 0 0 1 and Β = 4 0 1 0 are of the same type. They are square matrices of order 2. But they are not equal. For instance, the element in the second row and first column a 21 is equal to zero and is not equal to b 21 = 1.
- Brian H. Chirgwin, Charles Plumpton(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
CHAPTER V MATRICES 5:1 Introduction and notation In many of the applications of mathematics to physics, chemistry and engineering, and also in many branches of mathematics itself, sets of quantities occur which have similar meanings and which can naturally be set out in rectangular arrays. In Chapter I the set of direction cosines corresponding to a rotation of three dimensional axes formed such an array; the coefficients of a conic (2 dimensions) or of a quadric (3 di-mensions) can also be set out as symmetrical square arrays. In mechanics the coefficients of inertia (moments and products) of a rigid body can form a square array; rotations, angular velocity and angular momentum utilise sets of quantities which arise from the re-lations used in geometry for rotations of axes. The general equations of motion giving the small oscillations of a mechanical system about a position of equilibrium involve coefficients which are naturally arranged in square arrays. The development of this theory of small oscillations (see Vol. VI Chapter IX) is a particular use of the general theory of ma-trices which we discuss later in this chapter. Similarly, in electricity the equations governing the behaviour of networks of conductors carrying currents and the equations for systems of conductors in electrostatics involve similar arrays of coefficients. In the equations for networks carrying alternating currents these co-efficients may be complex numbers. A closer study of the mathematics of these phenomena shows that the arrays are formed of coefficients in sets of linear simultaneous equa-tions which arise in the theory, or the arrays are the coefficients in quadratic forms of several variables. A set of linear equations ^11 x l a 12 x 2 i ' ' * ~r a ±n x n ~ 1 » ft 21 ^1 T ^22 ^2 i ' * ' a 2n X n = *2 » (5 ) . . . , a ml X l + a m2 X 2 + + a mn x n = ^m 253 254 A COURSE OF MATHEMATICS involves the following three arrays.
eBook - PDF- Pease(Author)
- 1964(Publication Date)
- Academic Press(Publisher)
CHAPTER I Vectors and Matrices In this chapter, we shall lay the foundations of the rest of our work. We shall define what we mean by a vector (which is not the same thing, although it is a generalization, as the vectors one meets in, for example, vector analysis). We shall introduce the concept of a matrix as an operator that expresses a linear homogeneous relation between two vector spaces (a mapping of one vector space onto or into another). Finally, both as an example and because the problem is an important one, we shall briefly discuss the analysis of reciprocal ladder networks. 1. VECTORS The basic element that we shall consider is the vector. As we use the term, a vector is a generalization of the concept that we use in vector analysis or in elementary physics. It has some of the properties of a vector in physical 3-space, but not necessarily all of them. T o avoid confusion, we must take pains to know exactly what we mean by the term. We start with a partially abstract definition: Definition. A vector is a set of n numbers arranged in a de$nite To be completely general, we should say “a set of n numbers of The set of numbers so arranged are called the components of the order. a field,’ F,” or of symbols that stand for members of the field. vector. The number n is the dimensionality of the vector. A field is defined, roughly, as a collection of elements within which the operations of addition, subtraction, multiplication, and division, except by zero, are always possible and give a unique element of the field. The positive real numbers are not a field since subtraction is not always possible. Neither is the set of all integers, since division is not always possible. The two fields that will concern us are the field of all real numbers and the field of all complex numbers. The elements of the field being considered are called scalars. 1 2 I. VECTORS AND MATRICES We shall take a column as the standard form for a vector.
No longer available |Learn moreMathematical Methods for Physics
Using MATLAB and Maple
- J. R. Claycomb(Author)
- 2018(Publication Date)
- Mercury Learning and Information(Publisher)
Chapter 49 2 Chapter V ECTORS AND M ATRICES Chapter Outline 2.1 Vectors and Scalars in Physics 2.2 Matrices in Physics 2.3 Matrix Determinant and Inverse 2.4 Eigenvalues and Eigenvectors 2.5 Rotation Matrices 2.1 VECTORS AND SCALARS IN PHYSICS Physical quantities in nature can be scalars (such as mass, temperature, or density) or vectors (such as force or electric and magnetic fields). Both scalars and vectors are described by a numerical factor depending on the system of units used. Common units in physics are the MKS (meter, kilogram, second) and the CGS (centimeter, gram, second) systems. Vector quantities also have directionality. Two scalar quantities with the same units may be added or subtracted directly. Scalars with differing units may be multiplied or divided. For example, the density of an object is its mass divided by its volume. The directionality of vectors must be taken into account during addition, subtraction, and multiplication, however. 50 MATHEMATICAL METHODS FOR PHYSICS USING MATLAB AND MAPLE In this textbook, vectors are designated with boldfaced symbols such as A. Vectors with unit magnitude are usually boldfaced with a hat such as ˆ k . 2.1.1 Vector Addition and Unit Vectors Scalars are tensors of rank zero while vectors are tensors of rank one. A vector may be represented graphically by an arrow with a specified length (or magnitude) and a direction. Vectors with the same magnitude, length, and units are said to be equal. A vector translated in space remains unchanged unless it is rotated or stretched. Vectors may be added graphically by arranging them tip-to-tail in any order. The vector sum, known as the resultant, is found by constructing a vector from the tail of the first vector to the tip of the last vector. A vector A may be represented in Cartesian coordinates as ˆ ˆ ˆ x y z A A A = + + A i j k (2.1.1) where ˆ i , ˆ j , and ˆ k are unit vectors in the respective x-, y-, and z-directions.
eBook - PDFPhysical Oceanography
A Mathematical Introduction with MATLAB
- Reza Malek-Madani(Author)
- 2012(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 2 Matrix Algebra In this chapter we develop the basic concepts and tools in matrix algebra, including vector spaces and subspaces, systems of algebraic equations, determinants and inverse of matrices, Gaussian elimination, and eigen-values and eigenvectors. Each mathematical topic is supplemented with the elementary MATLAB functions that relate to it. 2.1 Vectors and Matrices A vector is a quantity that has magnitude and direction, while a scalar is a quantity with magnitude only. As is standard in science, scalars are often denoted by Latin or Greek letters, such as a , b , α and β , and vectors are displayed in boldface – x , y , and e 2 . Physical con-cepts such as force, velocity, and acceleration are represented by vectors, while quantities such as mass, pressure, temperature, and salinity are examples of scalars. We adopt the natural geometric interpretation of a vector v in the plane or the three-dimensional space, as an arrow that begins at the origin of the coordinate axes, is parallel to the direction of the vector, and has its length equal to the magnitude of the vector. In this setting, we use the coordinates of the endpoint of the arrow to identify the vector. For example v = angbracketleft 1 , -2 , 2 angbracketright is the vector that originates at (0 , 0 , 0) and ends up at (1 , -2 , 2). Note the use of angbracketleft and angbracketright to denote a vector, while ( and ) are used to denote coordinates of positions. With this interpretation in mind, the length or the magnitude of the vector v = angbracketleft a 1 ,a 2 ,a 3 angbracketright , denoted by || v || , is the distance from (0 , 0 , 0) to ( a 1 ,a 2 ,a 3 ): || v || = radicalBig a 2 1 + a 2 2 + a 2 3 . (2.1) Although vectors in physical settings typically have two or three com-41
eBook - PDFQuantum Computing
From Linear Algebra to Physical Realizations
- Mikio Nakahara, Tetsuo Ohmi(Authors)
- 2008(Publication Date)
- CRC Press(Publisher)
1 Basics of Vectors and Matrices The set of natural numbers { 1 , 2 , 3 , . . . } is denoted by N . The set of integers { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } is denoted by Z . Q denotes the set of rational num-bers. Finally R and C denote the sets of real numbers and complex numbers, respectively. Observe that N ⊂ Z ⊂ Q ⊂ R ⊂ C The vector spaces encountered in physics are mostly real vector spaces and complex vector spaces. Classical mechanics and electrodynamics are formu-lated mainly in real vector spaces while quantum mechanics (and hence this book) is founded on complex vector spaces. In the rest of this chapter, we briefly summarize vector spaces and matrices (linear maps), taking applica-tions to quantum mechanics into account. The Pauli matrices , also known as the spin matrices, are defined by σ x = 0 1 1 0 , σ y = 0 − i i 0 , σ z = 1 0 0 − 1 . They are also referred to as σ 1 , σ 2 and σ 3 , respectively. The symbol I n denotes the unit matrix of order n with ones on the di-agonal and zeros off the diagonal. The subscript n will be dropped when the dimension is clear from the context. The arrow → often indicates logical implication. We use e x and exp( x ) interchangeably to denote the exponential function. For any two matrices A and B of the same dimension, their commutator, or commutation relation, is a matrix defined as [ A, B ] ≡ AB − BA, while the anticommutator, or anticommutation relation, is { A, B } ≡ AB + BA. The symbol denotes the end of a proof. 3 4 QUANTUM COMPUTING 1.1 Vector Spaces Let K be a field, which is a set where ordinary addition, substraction, multi-plication and division are well-defined. The sets R and C are the only fields which we will be concerned with in this book. A vector space is a set where the addition of two vectors and a multiplication by an element of K , so-called a scalar , are defined. DEFINITION 1.1 A vector space V is a set with the following properties; (0-1) For any u, v ∈ V , their sum u + v ∈ V .
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