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Matrices
 

Matrices

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    Matrices Matrices Presentation Transcript

    • MATRICES
      María Isabel Cadena
      Métodos Numéricos
    • TYPES OF MATRICES
      UPPER TRIANGULAR MATRIX:
      The matrix A = (aij) a square matrix of order n. We say that A is upper triangular if all elements of A situated below the main diagonal are zero, ieaij = 0 for all i> j, i, j = 1 ,...., nFor example the matrices
    • LOWER TRIANGULAR MATRIX:
      The matrix A = (aij) a square matrix of order n. We say that A is lower triangular if all elements of A located above the main diagonal are zero, ieaij = 0 for all i <j, i, j = 1 ,...., nFor example, arrays
    • MATRIX TRANSPOSE:
      Given a matrix A, is called the matrix transpose of the matrix A is obtained by changing sort rows by the columns.
      • (At)t = A
      • (A + B)t = At + Bt
      • (α ·A)t = α· At
      • (A ·  B)t = Bt · At
    • Symmetric matrix
      A symmetric matrix is a square matrix that verifies:A = At.
      MATRIX INVERSE
      The product of a matrix by its inverse equals the identity matrix.A · A-1 = A-1 ° A = I
      PROPERTIES
      (A ° B) -1 = B-1 to-1(A-1) -1 = A(K • A) -1 = k-1 to-1(A t) -1 = (A -1) t
    • OPERATIONS WITH MATRICES
      SUM OF MATRICES:
      Given two matrices of the same size, A = (aij) and B = (bij) is defined as the matrix sum:
      A + B = (aij + bij).The matrix sum is obtained by adding the elements of the two arrays that occupy the same same position.
    • Properties of matrixaddition:
      • Internal:The sum of two matrices of order mxn matrix is another dimension mxn.
      • Associations:A + (B + C) = (A + B) + C
      • Neutral element:A + 0 = AWhere O is the zero matrix of the same dimension as matrix A.
      • Opposite element:A + (-A) = OThe matrix is opposite that in which all elements are changed in sign.
      • Commutative:A + B = B + A
    • Product of a scalar by a matrix:
      Given a matrix A = (aij) and a real number kR, defines the product of a real number by a matrix: the matrix of the same order as A, in which each element is multiplied by k.
      kA=(k aij)
    • Product Matrix:
      Two matrices A and B are multiplied if the number of columns of A matches the number of rows of B.Mm Mn x x n x m x p = M pThe element cij of the matrix product is obtained by multiplying each element in row i of matrix A for each element of column j of the matrix B and adding.
       
    • Product features matrix:
      • Associations:A ° (B ° C) = (A ° B) ° C
      • Neutral element:A · I = AWhere I is the identity matrix of the same order as the matrix A.
      • Not Commutative:A · B ≠ B * A
      • Product distributive over addition:A ° (B + C) = A ° B + A × C.
    • BIBLIOGRAPHY
      http://www.ditutor.com
      www.fisicanet.com