28_3 - Inverting Matrix AX = I X = A-1 How hard is the...

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1 Matrix inversion Inverting Matrix ± AX = I X = A -1 ± How hard is the matrix inversion problem? Theorem: Multiplication is as hard as inversion Proof: Let I(n) be the cost of inversion. Let M(n) be the cost of multiplication. Want to show I(n) = Θ (M(n)). ± M(n)=O(I(n)) ± I(n) = O(M(n)) Inverting Matrix ± AX = I X = A -1 ± How hard is the matrix inversion problem? Theorem: Multiplication is as hard as inversion Proof: Let I(n) be the cost of inversion. Let M(n) be the cost of multiplication. Want to show I(n) = Θ (M(n)). ± M(n)=O(I(n)) ± I(n) = O(M(n)) example ± Two other problems: ± Multiplication of two n-digit numbers x and y ± Squaring one n-digit number z ± Which problem is harder (assuming +/- can be done easily)? xy = ((x+y) 2 -(x-y) 2 )/4 Inverting Matrix ± M(n)=O(I(n)) Consider A, B, C s.t. AB = C Construct ± I(n) = O(M(n)) I DD I B I AB A I D I B I A I D n n n n n n = = = 1 1 0 0 0 0 0 0 0 , , . 3 assumming )), ( ( ) ( )
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28_3 - Inverting Matrix AX = I X = A-1 How hard is the...

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