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.