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!

Matrix equation

  1. Dec 26, 2007 #1
    We have [tex]X\in R^{s\times n}[/tex], [tex]A\in R^{n\times s}[/tex], [tex]I\in R^{s\times s}[/tex], where [tex]I[/tex] stands for the indentity matrix.
    Now if we assume that [tex]rank(A)=s[/tex], can we conclude that there must exist a solution [tex]X[/tex] to matrix equation [tex]XA=I[/tex]?

    For me the answer is obviously "yes" if we think this problem in the language of linear map instead of matrices.

    Let [tex]T[/tex] be the linear map whose matrix is [tex]A[/tex] with respect to the standard basis. Then the above question is converted into if [tex]T[/tex] is injective, can we always find an [tex]S[/tex] maping [tex]R^{n}[/tex] to [tex]R^{s}[/tex], such that [tex]ST[/tex] is an identity operator in [tex]R^{s}[/tex]?

    The linear map [tex]S[/tex] is defined as follows:
    for any [tex]x\in range(T)[/tex], [tex]Sx[/tex] is the unique vector [tex]y\in R^{s}[/tex] such that [tex]Ty=x[/tex]. this is ensured because [tex]T[/tex] is injective.
    for other [tex]x\in R^{n}[/tex], [tex]Sx=0[/tex].

    Though knowing for sure how [tex]S[/tex] behaves, is there any way to write out the matrix of [tex]S[/tex] with respect to the standard basis?
    i.e. what does the solution [tex]X[/tex] to matrix equation [tex]XA=I[/tex] look like?
     
  2. jcsd
  3. Dec 28, 2007 #2
    Use singular value decomposition,

    [tex]A=U\Sigma V^T \Longrightarrow X = V\hat{\Sigma}U^T[/tex]

    where [tex]\hat{\Sigma}[/tex] has the reciprocals of the original on the diagonal. Note that there is a technical peculiarity but I guess you can figure it out. Search for left inverse (or pseudo-inverseor Moore-Penrose inverse). It is almost fun .
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Matrix equation
  1. Matrix Equation (Replies: 8)

Loading...