How to solve this linear matrix equation











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I have an equation with one unknown matrix variable $mathbf{M}$:



$$
frac{1}{2}vec(mathbf{W})^T(mathbf{V} otimes mathbf{U})vec(mathbf{M}) + frac{1}{2}vec(mathbf{M})^T(mathbf{V} otimes mathbf{U})vec(mathbf{W})
$$
$$
= frac{1}{tau^2}vec(mathbf{W})^Tvec(mathbf{W}_0)+frac{1}{sigma^2}vec(mathbf{T})^Tvec(mathbf{WX})
$$



Where $otimes$ denotes the Kroenecker product.



I know that
$$
mathbf{V} otimes mathbf{U} = (frac{1}{tau^2}mathbf{I}_q + frac{1}{sigma^2}mathbf{X}mathbf{X}^T)otimes mathbf{I}_D
$$



And $mathbf{W}$ should somehow disappear out of the equation.
$mathbf{X}$, $mathbf{T}$, $mathbf{W}_0$, $tau$ and $sigma$ are known.



How can I solve for $mathbf{M}$ (ideally unvectorized $mathbf{M}$)?



ADDITIONAL INFORMATION: I know from my application that $mathbf{V} otimes mathbf{U}$ is symmetric.






linear-algebra







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    up vote
    0
    down vote

    favorite












    I have an equation with one unknown matrix variable $mathbf{M}$:



    $$
    frac{1}{2}vec(mathbf{W})^T(mathbf{V} otimes mathbf{U})vec(mathbf{M}) + frac{1}{2}vec(mathbf{M})^T(mathbf{V} otimes mathbf{U})vec(mathbf{W})
    $$
    $$
    = frac{1}{tau^2}vec(mathbf{W})^Tvec(mathbf{W}_0)+frac{1}{sigma^2}vec(mathbf{T})^Tvec(mathbf{WX})
    $$



    Where $otimes$ denotes the Kroenecker product.



    I know that
    $$
    mathbf{V} otimes mathbf{U} = (frac{1}{tau^2}mathbf{I}_q + frac{1}{sigma^2}mathbf{X}mathbf{X}^T)otimes mathbf{I}_D
    $$



    And $mathbf{W}$ should somehow disappear out of the equation.
    $mathbf{X}$, $mathbf{T}$, $mathbf{W}_0$, $tau$ and $sigma$ are known.



    How can I solve for $mathbf{M}$ (ideally unvectorized $mathbf{M}$)?



    ADDITIONAL INFORMATION: I know from my application that $mathbf{V} otimes mathbf{U}$ is symmetric.






    linear-algebra







    share|cite|improve this question


























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I have an equation with one unknown matrix variable $mathbf{M}$:



      $$
      frac{1}{2}vec(mathbf{W})^T(mathbf{V} otimes mathbf{U})vec(mathbf{M}) + frac{1}{2}vec(mathbf{M})^T(mathbf{V} otimes mathbf{U})vec(mathbf{W})
      $$
      $$
      = frac{1}{tau^2}vec(mathbf{W})^Tvec(mathbf{W}_0)+frac{1}{sigma^2}vec(mathbf{T})^Tvec(mathbf{WX})
      $$



      Where $otimes$ denotes the Kroenecker product.



      I know that
      $$
      mathbf{V} otimes mathbf{U} = (frac{1}{tau^2}mathbf{I}_q + frac{1}{sigma^2}mathbf{X}mathbf{X}^T)otimes mathbf{I}_D
      $$



      And $mathbf{W}$ should somehow disappear out of the equation.
      $mathbf{X}$, $mathbf{T}$, $mathbf{W}_0$, $tau$ and $sigma$ are known.



      How can I solve for $mathbf{M}$ (ideally unvectorized $mathbf{M}$)?



      ADDITIONAL INFORMATION: I know from my application that $mathbf{V} otimes mathbf{U}$ is symmetric.






      linear-algebra







      share|cite|improve this question















      I have an equation with one unknown matrix variable $mathbf{M}$:



      $$
      frac{1}{2}vec(mathbf{W})^T(mathbf{V} otimes mathbf{U})vec(mathbf{M}) + frac{1}{2}vec(mathbf{M})^T(mathbf{V} otimes mathbf{U})vec(mathbf{W})
      $$
      $$
      = frac{1}{tau^2}vec(mathbf{W})^Tvec(mathbf{W}_0)+frac{1}{sigma^2}vec(mathbf{T})^Tvec(mathbf{WX})
      $$



      Where $otimes$ denotes the Kroenecker product.



      I know that
      $$
      mathbf{V} otimes mathbf{U} = (frac{1}{tau^2}mathbf{I}_q + frac{1}{sigma^2}mathbf{X}mathbf{X}^T)otimes mathbf{I}_D
      $$



      And $mathbf{W}$ should somehow disappear out of the equation.
      $mathbf{X}$, $mathbf{T}$, $mathbf{W}_0$, $tau$ and $sigma$ are known.



      How can I solve for $mathbf{M}$ (ideally unvectorized $mathbf{M}$)?



      ADDITIONAL INFORMATION: I know from my application that $mathbf{V} otimes mathbf{U}$ is symmetric.






      linear-algebra




      linear-algebra






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








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