Results 1 to 4 of 4

Math Help - Non-square inverse proofs

  1. #1
    Newbie
    Joined
    Nov 2007
    Posts
    2

    Non-square inverse proofs

    I need to prove the following:

    1 A non-square matrix cannot have both a left and a right inverse
    2 If a non-square matrix has a left(right) inverse, it has infinitely many.
    3 If m<n, a non-square matrix has a right inverse if and only if rank A=m
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by needhelp32 View Post
    I need to prove the following:

    1 A non-square matrix cannot have both a left and a right inverse
    2 If a non-square matrix has a left(right) inverse, it has infinitely many.
    3 If m<n, a non-square matrix has a right inverse if and only if rank A=m
    It is always true that rank(AB)≤rank(B) (because the rows of AB are linear combinations of the rows of B). Also, rank(AB)≤rank(A) (because the columns of AB are linear combinations of the columns of A).

    So for example if A is an mn matrix with m<n, then rank(A)≤m, and rank(AB)≤m. Therefore AB can never be equal to the nn identity matrix, which has rank n.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2007
    Posts
    2
    Thank you very much. I've now figured out (1). Does anyone know how to prove (2) in general.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by needhelp32 View Post
    Does anyone know how to prove (2) in general.
    If A is mn with m<n, then rank(A) < n, so A has a nonzero kernel. Let x be a nonzero vector in this kernel, and let C be the matrix of any linear transformation whose range consists of multiples of x. Then AC=0. If B is a right inverse of A then so is B+λC, for any value of the scalar λ.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Divergence of inverse square force field
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 28th 2011, 11:41 PM
  2. How to calculate inverse square law?
    Posted in the Math Topics Forum
    Replies: 8
    Last Post: January 28th 2011, 03:19 PM
  3. Replies: 8
    Last Post: February 10th 2010, 11:44 PM
  4. How to find Inverse of a Non-Square Matrix...
    Posted in the Algebra Forum
    Replies: 1
    Last Post: December 11th 2009, 04:18 AM
  5. Square root's inverse right but restrictions?
    Posted in the Math Topics Forum
    Replies: 2
    Last Post: June 8th 2009, 12:47 AM

Search Tags


/mathhelpforum @mathhelpforum