Matrices

8 Key excerpts on "Matrices"

• 

  eBook - ePub

  3D Math Primer for Graphics and Game Development

  • Fletcher Dunn, Ian Parberry(Authors)
  • 2011(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  Chapter 4

  Introduction 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.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Advanced Engineering Mathematics with Mathematica

  • Edward B. Magrab(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  1 Matrices, Determinants, and Systems of Equations 1.1 Definitions A matrix is a rectangular array of numbers (or symbols or expressions) consisting of m rows and n columns. Such an array is called a matrix of order (m × n) [or simply an (m × n) matrix], and is denoted as A = a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ a m 1 a m 2 ⋯ a m n (1.1) where the a ij are the elements of the array. The first subscript denotes the row, and the second subscript denotes the column in which the element appears. When the number of rows equals the number of columns, that is, when m = n, the matrix is called a square matrix of order n. Operations with Matrices are subject to certain rules, which will be indicated as we progress. It should be realized that a matrix is not a single quantity; it is a representation of an array of entities. For a (2 × 3) matrix, Eq. (1.1) is created with Mathematica procedure M1.1. The transpose of a matrix is obtained by interchanging the rows and columns and the transpose operation is denoted with a superscript T. Thus, the transpose of the (m × n). matrix A is B = A T = a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ a m 1 a m 2 ⋯ a m n T = b 11 = a 11 b 12 = a 21 ⋯ b 1 m = a m 1 b 21 = a 12 b 22 = a 22 ⋯ b 2 m = a m 2 ⋮ b n 1 = a 1 n b[--=PLGO-SEPARATOR. =--]n 2 = a 2 n ⋯ b n m = a m n (1.2) where B is now an (n × m) matrix. For a (2 × 3) matrix, Eq. (1.2) is created with Mathematica procedure M1.2. A symmetric matrix is a square matrix in which a ij = a ji. Thus, for a symmetric matrix A = A T (1.3) A skew-symmetric matrix is a square matrix in which a ij = − a ji, which implies that a ii = 0. Thus, A = − A T A column matrix is a matrix with only one column, that is, an (m × 1) matrix. This matrix is denoted as a = a 11 a 21 ⋮ a m 1 = a 1 a 2 ⋮ a m (1.4) A column matrix is frequently referred to as a column vector. A row matrix is a matrix with only one row, that is, a (1 × m) matrix

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Mathematical Methods for Finance

  Tools for Asset and Risk Management

  • Sergio M. Focardi, Frank J. Fabozzi, Turan G. Bali(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 4 Matrix Algebra

  I n mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrix algebra generalizes classical analytical notions such as derivatives and exponentials to higher dimensions. Matrix algebra collects the various partial derivatives of a single function with respect to many variables, and of a multivariate function with respect to a single variable, into vectors and Matrices that can be treated as single entities. This greatly simplifies operations, such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. Calculations in portfolio theory, financial economics, and financial econometrics rely on the use of matrix algebra because of the need to manipulate large data inputs.

  • Matrix algebra is used for optimal portfolio selection.
  • Matrix algebra is useful for computing expected return of a portfolio that contains many assets.
  • Matrix algebra is useful for computing the variance (or risk) of a portfolio that contains many assets.
  • Optimal portfolio weights are calculated by maximizing the risk-adjusted return of a portfolio or by maximizing expected utility of a risk-averse investor. For either case, matrix algebra is useful for determining optimal asset allocation.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Mathematics for Economics and Finance

  • Michael Harrison, Patrick Waldron(Authors)
  • 2011(Publication Date)
  • Routledge

    (Publisher)

  matrix.

  Definition 1.2.1 A matrix is a rectangular array of scalars called elements (or entries). In this book, scalars will generally be real numbers.

  Such an array is exemplified in the following general notation for a matrix.

  Notation 1.2.1 A general matrix A can be written asm

  A =

  [

  a 11

  a 12

  ⋯

  a

  1 n

  a 21

  a 22

  ⋯

  a

  2 n

  ⋮ ⋮ ⋮

  a

  m 1

  a

  m 2

  ⋯

  a

  m n

  ]

  =

  [

  a

  i j

  ]

  Notice the use of a bold upper-case Roman letter for the matrix and the corresponding lower-case letter for its scalar elements. This will be our usual convention when referring to Matrices and scalars. Note also the use of subscripts denoting the row and the column to which each element belongs. The number of rows and the number of columns determine the order or dimension of the matrix; here the order is m × n, signifying m rows and n columns and hence mn elements in all. Sometimes it may be useful to make the order explicit by means of a subscript: A

  m×n

  . Note further that it is often very useful to denote a matrix by means of its typical element, [aij ]. Thus an efficient way of representing a matrix is to write [aij ]

  m×n

  . When m = n, and the number of rows and columns is the same, we say that the matrix is a square matrix.

  A matrix for which m = 1 is a single row of n elements; this is often called a row vector. Similarly, an m × 1 matrix may be called a column vector. We shall also sometimes refer to m × 1 column vectors as m-vectors, especially from Chapter 5 onwards. Our notation for row and column vectors, respectively, is as follows.

  Notation 1.2.2 A general row vector r can be written asm

  r =

  [

  r 1

  r 2

  ⋯

  r n

  ]

  Notation 1.2.3 A general column vector c can be written as

  c =

  [

  c 1

  c 2

  ⋮

  c m

  ]

  or (c1 ,c2 , ... ,c

  m

  ) to economize on space

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Elements of Linear Algebra

  • P.M. Cohn(Author)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  3 Matrices 3.1 DEFINITION OF A MATRIX There is yet another way of regarding an m × n system of equations, which will lead to an even shorter way of writing it than the vector notation used in Chapter 2. Let us first write the system as a 11 x 1 + ⋯ + a 1 n x n = y 1 ⋯ ⋯ ⋯ a m 1 x 1 + ⋯ + a m n x n = y m (3.1) where we have replaced the constants k i on the right by variables y i. If we take x = (x 1, …, x n) T and y = (y 1, …, y m) T to be column vectors, we may regard the set of coefficients (a ij) in (3.1) as an operator which acts on the n -vector x to produce the m -vector y. In this sense we write (3.1) as (a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋯ ⋯ a m 1 a m 2 ⋯ a m n) (x 1 x 2 ⋮ x n) = (y 1 y 2 ⋮ y m) (3.2) The rectangular array of. coefficients (a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋯ ⋯ a m 1 a m 2 ⋯ a m n) (3.3) is called a matrix with the entries a ij and the left-hand side of (3.2) may be regarded as the ‘product’ of this matrix by x. A comparison with (3.1) shows that this product is evaluated by multiplying the elements of the j th column of (3.3) by x j and adding the column vectors so obtained. The sum of these vectors has its i th component equal to a i 1 x 1 + … + a in x n (i = 1, …, m), and the vector equation in (3.2) states that this vector is equal to the column vector y = (y 1, …, y m) T, which is in fact a restatement of the equations (3.1). A matrix with m rows and n columns as in (3.3) is also called an m × n matrix, and the set of all m × n Matrices with entries in R will be denoted by m R n. When m = 1, this reduces to the vector space of rows 1 R n, also written R n, while n = 1 gives the space m R l of columns, also written m R, in agreement with our earlier notation R n for the space of rows and m R for the space of columns of the indicated dimensions. If we denote the matrix (3.3) by A, we may write (3.2) more briefly as A x = y In (3.2) we applied the matrix A to x and obtained the result y

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Matrices and Linear Transformations

  Second Edition

  • Charles G. Cullen(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  Definition 1.1 show that

  a)

  b)

  c)

  d)

      7. Show that the system (x ) (see Example 5 ) is indeed a field. Be sure to consider what “ +,” “ ·,” and “ = ” mean in this system.

  1.3

     Matrices

  The remainder of this chapter is concerned principally with Matrices and their relationship to systems of linear algebraic equations. The first order of business is to formally define the term matrix.

  Definition 1.3 A matrix over the field is a rectangular array of elements from . Two Matrices are equal if and only if they are identical.

  We will normally use capital italic letters to designate Matrices, for example:

  If a matrix has r rows and c columns, we will say that it is r by c (r × c ), the number of rows always being listed first. For the above Matrices N is 3 × 4, P is 2 × 2, and Q is 3 × 2.

  The element in the i th row and j th column of the matrix A (the i , j position of A ) will generally be denoted by

  aij

  and we will write A = (

  aij

  ) to indicate this. Note again that the row index is always listed first. In the above examples, n 23 = – 1, n 32 = 3, while p 12 = 0, q 21 = – 2.

  The usual way of describing a general matrix with r rows and c columns is

  One says that the matrix M above has order r by c (r × c ); by agreement the row size is always listed first. The double subscript notation for the elements permits us to discuss the elements or entries of a general matrix in an efficient manner. Note that the first subscript on the entry

  mij

  indicates the row in which the entry occurs while the second subscript indicates the column in which it occurs. The ordered pair (i , j ) is called the address of the entry

  mij

  , and the entry in the (i , j ) position of M is

  mij

  . The entry

  mij

  will be said to have row index i and column index j .

  If the matrix has only one row or one column, we will normally use only a single subscript to designate its elements. For example,

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Engineering Mathematics with Applications to Fire Engineering

  • Khalid Khan, Tony Lee Graham(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  4 Determinants and Matrices

  4.1 Background

  Matrices with different dimensions are a standard method for solving a wide range of problems in many disciplines. In science and engineering, when considering particular problems, the derivation of the solutions to these problems can come down to solving a system of linear equations. In this chapter, different methods for solving a linear system of equations will be discussed and these methods involve the concepts of determinants and Matrices.

  4.2 Introduction to Determinants

  Determinants arise naturally in the solution of a set of linear equations. Also they will help solve linear equations using the matrix inversion method (see Section 4.3.6 ). Starting with a general set of two linear simultaneous equations as follows:

  a x + b y = e

  4.1

  4.1

  c x + d y = f

  4.2

  4.2

  where a , b , c , d , e , and f are constants. How do you find the values of x and y ? Since the values of a , b , c , and d are not known, all that can be done is try to eliminate either x or y .

  If one decides to get rid of y , then to make the coefficients of y the same, first multiply Equation 4.1 by d and Equation 4.2 by b :

  a d x + b d y = d e

  4.3

  4.3

  b c x + b d y = b f

  4.4

  4.4

  Then subtracting Equation 4.4 from Equation 4.3 :

  a d x − b c x = d e − b f   ∴   x ( a d − b c ) = d e − b f   ∴   x =

  d e − b f

  a d − b c

  A similar approach, getting rid of x , would give the result as

  y =

  a f − c e

  a d − b c

  Now provided that the denominator term ad − bc ≠ 0, then a symbol can be used to define a 2-by-2 determinant using parallel lines (∣ ∣), as shown next:

  |

  a b

  c d

  |

  ≜ a d − b c

  4.5

  4.5

  Note:The symbol ≜ means “defined as.”

  This definition given by Equation 4.5 is just the difference in product of the diagonals.

  Notice also how these answers for x and y

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Linear Algebra with Applications

  • Roger Baker, Kenneth Kuttler(Authors)
  • 2014(Publication Date)
  • WSPC

    (Publisher)

  12 = 2, etc.

  We shall often need to discuss the tuples which occur as rows and columns of a given matrix A. When you see

  (a1 ··· a

  p

  ),

  this equation tells you that A has p columns and that column j is written as a

  j

  . Similarly, if you see

  this equation reveals that A has q rows and that row i is written r

  i

  .

  For example, if we could write and

  where r1 = (1 2), r2 = (3 2), and r3 = (1 −2).

  There are various operations which are done on Matrices. Matrices can be added, multiplied by a scalar, and multiplied by other Matrices. To illustrate scalar multiplication, consider the following example in which a matrix is being multiplied by the scalar, 3.

  The new matrix is obtained by multiplying every entry of the original matrix by the given scalar. If A is an m × n matrix, −A is defined to equal (−1) A.

  Two Matrices must be the same size to be added. The sum of two Matrices is a matrix which is obtained by adding the corresponding entries. Thus Two Matrices are equal exactly when they are the same size and the corresponding entries are identical. Thus

  because they are different sizes. As noted above, you write (cij ) for the matrix C whose ijth entry is cij . In doing arithmetic with Matrices you must define what happens in terms of the cij , sometimes called the entries of the matrix or the components

  Sign up to read

  Learn more about book