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Lecture23aa - Lecture 1 Multiplication of Two Matrices 2...

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Lecture 3/14/08 1. Multiplication of Two Matrices 2. Converting LP Problem to Standard Form a. Slack variables b. Surplus variables 3. Standard Form LP Written in Matrix Notation
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Multiplication of Two Matrices Multiplication of a row vector with a column vector Let A=( a ij ) 1 x n be an n-row vector [ ] n a a a A 1 12 11 ... = B =( b ij ) n x 1 be an n-column vector = 1 21 11 ... n b b b B Then the product of A and B = = n j j j b a 1 1 1 1 1 21 12 11 11 ... n n b a b a b a + + + =
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Example:Product of a Row Vector with Column Vector [ ] 3 4 = A = 3 2 B A*B = 4*2 + 3*3 =17
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Product of A=( a ij ) mxk and B=(b ij ) kxn Product is an m x n matrix C = (c ij ) mxn Cij = Product of i th row of A and j th column of B # of columns of A = # of rows of B
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Matrix Multiplication Example = 2 2 3 4 A 2 2 x = 5 3 1 2 B 2 2 x A*B = C 2x2 = 22 21 12 11 c c c c [ ] = 3 2 3 4 c 11 =4*2+3*3=17 [ ] = 5 1 3 4 c 12 =4*1+3*5=19 c 21 =10 c 22 =12 = 12 10 19 17
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Properties of Matrix Multiplication 1. In most cases AB BA 2. Matrix multiplication is associative: A=( a ij ) jxk , B=(b ij ) kxm , and C = (c ij ) mxn Then A(BC)=(AB)C 3. If A is an nxn square matrix and I nxn is nxn identity matrix: then AI = IA = A
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LP Problem in Standard Form An equivalent LP where 1. All constraints are equations 2. All variables are non-negative
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Slack Variables To convert a constaint to an equality: 1. A slack variable is added to the left hand side of each constaint: a. Denoted by s i for the ith constraint.
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