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16-Matrix-Algebra-4UP

16-Matrix-Algebra-4UP - Definitions Matrix algebra MATLAB...

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Unformatted text preview: Definitions Matrix algebra MATLAB specifics Matrix Algebra Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Matrix Algebra MATH 2070U 1 / 23 Definitions Matrix algebra MATLAB specifics Matrix Algebra 1 Basic definitions: matrices, vectors, scalars 2 Algebraic operations on matrices 3 MATLAB specific operators for matrix/array computations c D. Aruliah (UOIT) Matrix Algebra MATH 2070U 2 / 23 Definitions Matrix algebra MATLAB specifics 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 c D. Aruliah (UOIT) Matrix Algebra MATH 2070U 4 / 23 Definitions Matrix algebra MATLAB specifics Vectors n-vector : “skinny” matrix (dimension n × 1 or 1 × n ) 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 c D. Aruliah (UOIT) Matrix Algebra MATH 2070U 5 / 23 Definitions Matrix algebra MATLAB specifics Vectors and matrices in M ATLAB Columns separated by , (optional), rows by ; Indexing: A 2,3 7→ A ( 2 , 3 ) , x 3 7→ x ( 3 ) or x ( 3 , 1 ) or x ( 1 , 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 matrices/vectors returned by function size A = [ 1 2 3; 4 5 6 ]; size(A), size(A,1), size(A,2) c D. Aruliah (UOIT) Matrix Algebra MATH 2070U 6 / 23 Definitions Matrix algebra MATLAB specifics Zero and identity matrices Zero matrix 0 ∈ R m × n ∀ A ∈ R m × n A + = + A = A , where 0 = ··· ··· ....
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