Matrix-Algebra - Matrices Operations Linear Systems Special...

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Matrices Operations Linear Systems Special Matrices Matrix Algebra Dhavide Aruliah Faculty of Science MATH 2070/2072 Matrices Operations Linear Systems Special Matrices Outline Matrices, Vectors, & Scalars Operations with Matrices Systems of Linear Equations Special Matrix Families
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Matrices Operations Linear Systems Special Matrices Matrices Matrix A R m × n is rectangular array of numbers A = a 1,1 a 1,2 · · · a 1, n - 1 a 1, n a 2,1 a 2,2 · · · a 2, n - 1 a 2, n . . . . . . . . . . . . a m - 1,1 a m - 1,2 · · · a m - 1, n - 1 a m - 1, n a m ,1 a m ,2 · · · a m , n - 1 a m , n Numbers a i , j = elements of A = entries of A First index i of element a i , j = row index Second index j of element a i , j = column index Matrices Operations Linear Systems Special Matrices Vectors Special case of a “skinny” matrix n -vector is matrix x of dimension n × 1 x = x 1 x 2 . . . x n - 1 x n or x T = ( x 1 x 2 · · · x n - 1 x n ) Elements x i = components of x Convention: vectors generically column vectors assume x R n means x R n × 1 Scalars are vectors of length 1/matrices of dimension 1 × 1
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Matrices Operations Linear Systems Special Matrices Vectors and Matrices in M ATLAB Columns separated by , (optional), rows by ; Indexing: A 2,3 A ( 2 , 3 ) Colon indexing: A : ,2 means 2 nd column of A A 3,: means 3 rd row of A A 1: 3,2: 5 means 1 st –3 rd rows, 2 nd –5 th columns of A Dimensions of matrix A returned by size >> A = [ 1 2 3; 4 5 6 ]; >> size(A) >> size(A,1) >> size(A,2) Matrices Operations Linear Systems Special Matrices Matrix Addition If A R m × n , B R m × n , C = A + B R m × n has elements c i , j = a i , j + b i , j ( 1 i m , 1 j n ) Matrix addition in M ATLAB uses operator + >> A = [ 1 2 3; 4 5 6]; >> B = [ 7 8 9; 10 11 12 ]; >> C = A + B Matrices must be conformable (same shape) for addition >> a = [1,2,3]; >> b = [4;5;6]; >> a+b % Common mistake!
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Matrices Operations Linear Systems Special Matrices Scalar Multiplication If μ R and A R m × n , C = μ A R m × n has elements c i , j = μ a i , j ( 1 i m , 1 j n ) Scalar multiplication in M ATLAB uses operator * >> A = [ 1 2 3; 4 5 6 ]; >> mu = 0.5; >> C = mu * A Matrices Operations Linear Systems Special Matrices Matrix Multiplication If A R m × s , B R s × n , matrix product C = AB R m × n has elements c i , j = s k = 1 a i , k b k , j ( 1 i m , 1
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