Romberg and M Davenport Last updated 333 Technical Details

# Romberg and m davenport last updated 333 technical

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Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 3:33, November 20, 2019

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Technical Details: Matrix Inversion Lemma The general statement of the Sherman-Morrison-Woodbury iden- tity is that ( W + X T Y Z ) - 1 = W - 1 - W - 1 X T ( Y - 1 + ZW - 1 X T ) - 1 ZW - 1 where W is N × N and invertible, X and Z are R × N , and Y is R × R and invertible. The proof of this is straightforward. Given any right hand side v R N , we would like to solve ( W + X T Y Z ) w = v (1) for w . Set z = Y Zw Y - 1 z = Zw . We now have the set of two equations W w + X T z = v Zw - Y - 1 z = 0 . Manipulating the first equation yields w = W - 1 ( v - X T z ) , (2) and then plugging this into the second equation gives us ZW - 1 v - ZW - 1 X T z - Y - 1 z = 0 z = ( Y - 1 + ZW - 1 X T ) - 1 ZW - 1 v . (3) So then given any v R N , we can solve for w in ( 1 ) by combining ( 2 ) and ( 3 ) to get w = W - 1 v - W - 1 X T ( Y - 1 + ZW - 1 X T ) - 1 ZW - 1 v . As this holds for any right-hand side v , this establishes the result. 53 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 3:33, November 20, 2019
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