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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
asked 2 days ago
Sandi
1689
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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
asked 2 days ago
Sandi
1689
add a comment |
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
asked 2 days ago
Sandi
1689
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
asked 2 days ago
Sandi
1689
asked 2 days ago
Sandi
1689
edited yesterday
asked 2 days ago
Sandi
1689
asked 2 days ago
Sandi
1689
asked 2 days ago
Sandi
1689
1689
add a comment |
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