1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Singular Value Decomposition

  1. Sep 22, 2011 #1
    1. The problem statement, all variables and given/known data
    Hi

    If we have a matrix M, we can always make a singular value decomposition. If the matrix has full column rank (= is invertible), then the singular values are all nonzero, otherwise they are not all nonzero.

    Now, we can also associate a condition number to a matrix given by

    cond(M) = s1/sk

    where k is min(m, n) (where M is a m times n matrix). If M is rectangular, then it is not invertible, so the condition number should be infinite.

    Now, when I find the condition number of a rectangular matrix in MatLAB, it gives me a (large) number, which is finite. How can that be?
     
  2. jcsd
  3. Sep 22, 2011 #2

    hotvette

    User Avatar
    Homework Helper

    The reason is numerical roundoff error. If you repeat the exercise with random values in the matrix, the last singular value will sometimes be zero but very very small most of the time. For this reason some commerical programs allow the user to specific a threshold for what is considered zero.

    Also, I think the statement about rectangular matrices needs to be clarifed. If the matrix has more rows than columns and is full rank it won't have a zero singular value. If the number of rows is less than the number of columns it will have a zero singular value.
     
    Last edited: Sep 22, 2011
  4. Sep 22, 2011 #3
    Do you know a good reference on SVD? I didn't know the last part you stated, becaus it isn't in my book.
     
  5. Sep 22, 2011 #4

    hotvette

    User Avatar
    Homework Helper

    Not really, but the following link (see page 3) shows an example of a rectangular matrix with all non zero singular values.

    http://www.farinhansford.com/books/pla/material/pseudo.pdf [Broken]
     
    Last edited by a moderator: May 5, 2017
  6. Jun 15, 2012 #5
    Gilbert Strang, Linear Algebra.
    I learned a ton from it.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Singular Value Decomposition
Loading...