Romberg and M Davenport Last updated 333 Technical Details

Romberg and m davenport last updated 333 technical

This preview shows page 6 - 7 out of 7 pages.

Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 3:33, November 20, 2019
Image of page 6

Subscribe to view the full document.

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
Image of page 7
  • Fall '08
  • Staff

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes