How to solve this linear matrix equation












1














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



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



where $otimes$ denotes the Kroenecker product.



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



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 matrix-equations







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    1














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



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



    where $otimes$ denotes the Kroenecker product.



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



    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 matrix-equations







    share|cite|improve this question



























      1












      1








      1







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



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



      where $otimes$ denotes the Kroenecker product.



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



      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 matrix-equations







      share|cite|improve this question















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



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



      where $otimes$ denotes the Kroenecker product.



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



      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 matrix-equations




      linear-algebra matrix-equations






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 26 at 22:35









      Davide Giraudo

      125k16150259




      125k16150259










      asked Nov 18 at 14:58









      Sandi

      18811




      18811



























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