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Use SVD to show rank(XGY) = rank (G)

  1. Nov 7, 2012 #1
    1. Use the Singular Value Decomposition (SVD) of G to prove:
    [tex] rank(XGY^T) = rank (G) [/tex]
    Given that [itex]X[/itex] and [itex]Y[/itex] are two full column-rank matrices, but may not have the same rank.

    2. The attempt at a solution
    [tex]
    \begin{eqnarray*}
    XGY^T & = & X(U\Sigma V^T)Y^T \\
    & = & XU \left( \begin{array}{cc}
    \Sigma_{r} & 0 \\
    0 & 0 \\
    \end{array} \right) V^{T}Y^T
    \end{eqnarray*}
    [/tex]
    Now, [itex]XU[/itex] and [itex](VY)^T[/itex] are orthogonal matrices, because [itex]X[/itex] and [itex]Y[/itex] are orthogonal since they have full column rank (right?). Then somehow I want to argue that the rank of this matrix must the dimension of [itex]\Sigma_r[/itex]...
     
  2. jcsd
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