Results 1 to 4 of 4

Thread: 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
    10
    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
    10
    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: Mar 28th 2011, 11:41 PM
  2. How to calculate inverse square law?
    Posted in the Math Topics Forum
    Replies: 8
    Last Post: Jan 28th 2011, 03:19 PM
  3. Replies: 8
    Last Post: Feb 10th 2010, 11:44 PM
  4. How to find Inverse of a Non-Square Matrix...
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Dec 11th 2009, 04:18 AM
  5. Square root's inverse right but restrictions?
    Posted in the Math Topics Forum
    Replies: 2
    Last Post: Jun 8th 2009, 12:47 AM

Search Tags


/mathhelpforum @mathhelpforum