The use of prime when referring to row vectors a

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Unformatted text preview: he same elements in corresponding locations locations A = B ⇔ aij = bij ∀i, j Symbol ∀ means “for each and every”, ⇔means “if and only if” 4 2 3 4 = 2 0 3 2 ≠ 4 0 0 3 x 7 y = 4 8 Operations on Matrices Addition and subtraction Multiplication • Scalar multiplication • Multiplication of matrices Transposition Inversion 9 Addition and Subtraction of Matrices Two matrices can be added ⇔ two matrices are of the same dimension. In this case we say the two matrices are conformable for addition. In conformable Addition rule: add each pair of corresponding elements B = [b ], i = 1..m, j = 1..n C = A + B = [ a + b ], i = 1..m, j = 1..n A = aij , i = 1..m, j = 1..n ij ij 4 9 2 0 4 + 2 9 + 0 6 9 2 1 + 0 7 = 2 + 0 1 + 7 = 2 8 ij Subtraction rule: subtract each pair of corresponding elements 19 3 6 8 19 − 6 3 − 8 13 − 5 2 0 − 1 3 = 2 − 1 0 − 3 = 1 − 3 C = A −B 10 Scalar Multiplication Scalar multiplication of a matrix refers to multiplying each element of a Scalar each matrix by a (real) number. Number in this context is referred to as scalar. matrix 3 − 1 21 − 7 7 = 0 35 0 5 11 Conformability for Multiplication Conformability condition for multiplication: matrix A of dimension m x n is Conformability conformable for multiplication with matrix B of dimension k x l if n=k. conformable Translation: you can multiply A by B if the number of columns in A is equal Translation: to the number of rows in B. to The resulting dimension will be m x l Translation: matrix AxB will have A’s number of rows and B’s number of Translation: columns. columns. a c b d 1 Is conformable for multiplication with Is 3 2 4 since the number of columns in A (2) is equal to the since number of rows in B (also 2). number 12 Special Case of Matrix Multiplication: an Inner Product of Two Vectors Recap: vectors are special matrices one of whose dimensions is equal to 1 vectors (i.e. vector columns or row vectors). (i.e. Definition: an inner product of two vectors u and v with the equal number of elements is defined as u ⋅ v = u1v1 + u2 v2 + ... + un vn of where u = (u , u ,..., u ) where 1 2 n v = ( v1 , v2 ,..., vn ) Suppose I went shopping and bought 2 loaves of bread at $5 each and 3 packs of potato chips at $2 each. My consumption vector is: Q' = [ 2,3] My price vectors is: P ' = [ $5,$2] My shopping bill today will be exactly the inner product of P ' and Q ' My Q'⋅P ' = Q1 P + Q2 P2 = 2 × $5 + 3 × $2 = $16 1 13 Matrix Multiplication Rule Each element cij of C=AB is defined as an inner product of row Each matrix A and column j in matrix B. 1 3 A = 2 8 4 0 , B = 5 9 1× 5 + 3 × 9 32 C = AB = 2 × 5 + 8 × 9 = 82 4 × 5 + 0 × 9 20 i in 1. A and B are conformable for and multiplication: A’s number of columns is the same with B’s number of rows is 2. The resulting C=AB has A’s number of 2. rows (3) and B’s number of columns (1), so C is a column vector. so c11 = a11b11 + a12b21 C’s element in row 1 and column 1 is C’s an inner pro...
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