# The use of prime when referring to row vectors a

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online