Identify this type of matrix system of equation: $M_i = C_i + sum_{k=1}^K A_i M_k A_i^{prime}$
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I'm trying to find more information on how to solve this linear system of equations with a funky form.
$M_i = C_i + sum_{k=1}^K pi_k A_i M_k A_i^{prime}$,
for all $i in {1,...,K}$ s.t. $M_i$ is symmetric and PSD, and $sum_{k=1}^K pi_k =1$. $M_i$ is a covariance matrix. $M_i$, $A_i$, and $C_i$ are all $(n times n)$ matrices, and $pi_i$ is a scalar.
I've found on my own that this may be a special type of a DARE equation but the Wikipedia article was hard to understand.
I've found a way to write the equation in block-diagonal form, but I still can't solve it analytically, due to the off-diagonal elements of the first-term after the equality being a function of $M_1$ and $M_2$. Here it is for the case where $K=2$.
$begin{bmatrix} M_1 & 0 \ 0 & M_2 end{bmatrix} = begin{bmatrix} C_1 & -pi_1 A_1M_1A_2^{prime} - pi_2 A_1 M_2 A_2^{prime} \ -pi_1 A_2M_1A_1^{prime} - pi_2 A_2 M_2 A_1^{prime} & C_2 end{bmatrix} + begin{bmatrix} pi_1 A_1 & pi_2 A_1 \ pi_1 A_2 & pi_2 A_2 end{bmatrix} begin{bmatrix} M_1 & 0 \ 0 & M_2 end{bmatrix} begin{bmatrix} A_1^{prime} & A_2^{prime} \ A_1^{prime} & A_2^{prime} end{bmatrix} $
statistics systems-of-equations matrix-equations
asked Dec 5 at 1:33
hipHopMetropolisHastings
1718
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I'm trying to find more information on how to solve this linear system of equations with a funky form.
$M_i = C_i + sum_{k=1}^K pi_k A_i M_k A_i^{prime}$,
for all $i in {1,...,K}$ s.t. $M_i$ is symmetric and PSD, and $sum_{k=1}^K pi_k =1$. $M_i$ is a covariance matrix. $M_i$, $A_i$, and $C_i$ are all $(n times n)$ matrices, and $pi_i$ is a scalar.
I've found on my own that this may be a special type of a DARE equation but the Wikipedia article was hard to understand.
I've found a way to write the equation in block-diagonal form, but I still can't solve it analytically, due to the off-diagonal elements of the first-term after the equality being a function of $M_1$ and $M_2$. Here it is for the case where $K=2$.
$begin{bmatrix} M_1 & 0 \ 0 & M_2 end{bmatrix} = begin{bmatrix} C_1 & -pi_1 A_1M_1A_2^{prime} - pi_2 A_1 M_2 A_2^{prime} \ -pi_1 A_2M_1A_1^{prime} - pi_2 A_2 M_2 A_1^{prime} & C_2 end{bmatrix} + begin{bmatrix} pi_1 A_1 & pi_2 A_1 \ pi_1 A_2 & pi_2 A_2 end{bmatrix} begin{bmatrix} M_1 & 0 \ 0 & M_2 end{bmatrix} begin{bmatrix} A_1^{prime} & A_2^{prime} \ A_1^{prime} & A_2^{prime} end{bmatrix} $
statistics systems-of-equations matrix-equations
asked Dec 5 at 1:33
hipHopMetropolisHastings
1718
Which are your known variables and unknown variables?
– rzch
Dec 5 at 12:23
add a comment |
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down vote
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up vote
0
down vote
favorite
I'm trying to find more information on how to solve this linear system of equations with a funky form.
$M_i = C_i + sum_{k=1}^K pi_k A_i M_k A_i^{prime}$,
for all $i in {1,...,K}$ s.t. $M_i$ is symmetric and PSD, and $sum_{k=1}^K pi_k =1$. $M_i$ is a covariance matrix. $M_i$, $A_i$, and $C_i$ are all $(n times n)$ matrices, and $pi_i$ is a scalar.
I've found on my own that this may be a special type of a DARE equation but the Wikipedia article was hard to understand.
I've found a way to write the equation in block-diagonal form, but I still can't solve it analytically, due to the off-diagonal elements of the first-term after the equality being a function of $M_1$ and $M_2$. Here it is for the case where $K=2$.
$begin{bmatrix} M_1 & 0 \ 0 & M_2 end{bmatrix} = begin{bmatrix} C_1 & -pi_1 A_1M_1A_2^{prime} - pi_2 A_1 M_2 A_2^{prime} \ -pi_1 A_2M_1A_1^{prime} - pi_2 A_2 M_2 A_1^{prime} & C_2 end{bmatrix} + begin{bmatrix} pi_1 A_1 & pi_2 A_1 \ pi_1 A_2 & pi_2 A_2 end{bmatrix} begin{bmatrix} M_1 & 0 \ 0 & M_2 end{bmatrix} begin{bmatrix} A_1^{prime} & A_2^{prime} \ A_1^{prime} & A_2^{prime} end{bmatrix} $
statistics systems-of-equations matrix-equations
asked Dec 5 at 1:33
hipHopMetropolisHastings
1718
I'm trying to find more information on how to solve this linear system of equations with a funky form.
$M_i = C_i + sum_{k=1}^K pi_k A_i M_k A_i^{prime}$,
for all $i in {1,...,K}$ s.t. $M_i$ is symmetric and PSD, and $sum_{k=1}^K pi_k =1$. $M_i$ is a covariance matrix. $M_i$, $A_i$, and $C_i$ are all $(n times n)$ matrices, and $pi_i$ is a scalar.
I've found on my own that this may be a special type of a DARE equation but the Wikipedia article was hard to understand.
I've found a way to write the equation in block-diagonal form, but I still can't solve it analytically, due to the off-diagonal elements of the first-term after the equality being a function of $M_1$ and $M_2$. Here it is for the case where $K=2$.
$begin{bmatrix} M_1 & 0 \ 0 & M_2 end{bmatrix} = begin{bmatrix} C_1 & -pi_1 A_1M_1A_2^{prime} - pi_2 A_1 M_2 A_2^{prime} \ -pi_1 A_2M_1A_1^{prime} - pi_2 A_2 M_2 A_1^{prime} & C_2 end{bmatrix} + begin{bmatrix} pi_1 A_1 & pi_2 A_1 \ pi_1 A_2 & pi_2 A_2 end{bmatrix} begin{bmatrix} M_1 & 0 \ 0 & M_2 end{bmatrix} begin{bmatrix} A_1^{prime} & A_2^{prime} \ A_1^{prime} & A_2^{prime} end{bmatrix} $
statistics systems-of-equations matrix-equations
statistics systems-of-equations matrix-equations
asked Dec 5 at 1:33
hipHopMetropolisHastings
1718
asked Dec 5 at 1:33
hipHopMetropolisHastings
1718
asked Dec 5 at 1:33
hipHopMetropolisHastings
1718
asked Dec 5 at 1:33
hipHopMetropolisHastings
1718
asked Dec 5 at 1:33
hipHopMetropolisHastings
1718
1718
Which are your known variables and unknown variables?
– rzch
Dec 5 at 12:23
add a comment |
Which are your known variables and unknown variables?
– rzch
Dec 5 at 12:23
Which are your known variables and unknown variables?
– rzch
Dec 5 at 12:23
Which are your known variables and unknown variables?
– rzch
Dec 5 at 12:23
add a comment |
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Which are your known variables and unknown variables?
– rzch
Dec 5 at 12:23