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    Numerical Analysis II Essentials
    Numerical Analysis II Essentials
    Numerical Analysis II Essentials
    Ebook146 pages40 minutes

    Numerical Analysis II Essentials

    By The Editors of REA

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    About this ebook

    REA’s Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Numerical Analysis II covers simultaneous linear systems and matrix methods, differential equations, Fourier transformations, partial differential equations, and Monte Carlo methods.
    LanguageEnglish
    PublisherResearch & Education Association
    Release dateJan 1, 2013
    ISBN9780738672540
    Numerical Analysis II Essentials

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      Book preview

      Numerical Analysis II Essentials - The Editors of REA

      METHODS

      CHAPTER 9

      SIMULTANEOUS LINEAR SYSTEMS AND MATRIX METHODS

      9.1 MATRIX MATHEMATICS

      Definition 9.1.1

      Given two matrices B and C , then B’ is equal to C if both B and C are of order m × n, and the entries bij = cij ∀ i = 1,2, . . . ,m, j = 1,2, . . . ,n.

      Definition 9.1.2

      If B and C are m × n matrices, then the sum B + C = D is a matrix D of order m × n with entries dij = aij + bij ∀i = 1,2, . . . ,m j = 1,2, . . . ,n.

      Definition 9.1.3

      If B is a matrix of order m × n, and α any real number, then the product αB = B′ is an m × n matrix with entires bij = αbij ∀ i = 1,2, . . . ,m j = 1,2, . . . ,n.

      Let Bm+n denote a matrix of order m × n. All matrices discussed are from the field of real numbers.

      Definition 9.1.4

      If Bm×n , Cs×p are two matrices, then the product BC is defined if and only if n = s. The new product BC is of order m × p . If Bm×n, Cn×p , are two matrices, then BC = D is a matrix with entries dij given by

      In general, matrix multiplication is not commutative.

      Theorem 9.1.1 1

      If A, B, C are matrices that are suitable for addition, with α, β arbitrary real scalars, then properties (A) through (J) are sufficient to establish that all m × n real matrices form a vector space over ℝ

      A + B = B + A

      A + (B + C) = (A + B) + C

      Let Z be the identity element for matrix addition, A + Z = Z + A = A

      A + (−A) = − A + A = Z.

      k ∈ ℝ, A ∈ M then kA ∈ M

      α (A + B ) = αA + αB

      (α + β)A = αA + βA

      α(βA) = (αβ)A

      1A = A

      ZA = Z

      Definition 9.1.5

      The identity matrix of order n is denoted

      Theorem 9.1.2

      If Am×n , Bn×k, Ck×j ,Dn×k are matrices, α any real scalar, then

      A(BC) = (AB)C

      A(B + D) = AB + AD

      InB = B, BIk= B

      α(AB) = (αA)B = A(αB)

      Definition 9.1.6

      Any square matrix An×n is non-singular if there exist an n × n matrix denoted by A−1 such that A−1 A = AA−1 = In. A−1 is called the inverse of A.

      A is singular if A−1 does not exist, equivalently det(A) = 0. Only square matrices can have inverses.

      Theorem 9.1.3

      If A and B are non-singular matrices , then AB is also non-singular, and (AB)−1 = B−1 A−1. If A is a 2 × 2 matrix such that

      Then

      Theorem 9.1.4

      If A is a partitioned matrix such that

      then

      Theorem 9.1.5

      If a particular sequence of elementary row operations reduces A to I, then the same sequence reduces [A/I] to [I/A−1] .

      Theorem 9.1.6

      If a particular sequence of elementary column operations reduces A to I , then the same sequence reduces

      Theorem 9.1.7

      If

      is non-singular, and A11 is square and non-singular then

      where

      Determinants

      Definition 9.1.7

      If A is the matrix, [a] , then

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