Find nontrivial solution of generalized system of linear equations
$begingroup$
Given the ordinary matrix $X_{ij}$ of Dimension $n$ and some constant Matrices $A_{ij},B_{ij}$, find a solution of the System of equations
$X_{ij} = sum_{k,l=1}^n A_{ik}X_{kl}B_{lj}$
for all rows $i in { 1,dots,n }$ and columns $j in { 1,dots,n }$ that is different from Zero. Rewriting this equation as
$sum_{k,l=1}^n(delta_{ik}delta_{jl}-A_{ik}B_{lj})X_{kl} = 0$
and supposing that all Matrix entries can be expressed as a Vector of Dimension $n^2$, I can write
$Mvec X = 0$
where $(vec{X})_q = X_{ceiling(q/n),q (mod) n}$, $dim vec X = n^2$ is the vector-reformulated unknown Matrix and $M$ a constant Matrix of coefficients that are obtained by similar procedure making a rank 4 Matrix to a rank 2 Matrix. After this, I can use the Theorem that if $det M = 0$ then nontrivial Solutions exists. Is this correct?
Are there other ways to determine whether a nontrivial solution exist for this system?
linear-algebra systems-of-equations
asked Jan 17 at 12:31
kryomaximkryomaxim
2,3621825
$endgroup$
add a comment |
$begingroup$
Given the ordinary matrix $X_{ij}$ of Dimension $n$ and some constant Matrices $A_{ij},B_{ij}$, find a solution of the System of equations
$X_{ij} = sum_{k,l=1}^n A_{ik}X_{kl}B_{lj}$
for all rows $i in { 1,dots,n }$ and columns $j in { 1,dots,n }$ that is different from Zero. Rewriting this equation as
$sum_{k,l=1}^n(delta_{ik}delta_{jl}-A_{ik}B_{lj})X_{kl} = 0$
and supposing that all Matrix entries can be expressed as a Vector of Dimension $n^2$, I can write
$Mvec X = 0$
where $(vec{X})_q = X_{ceiling(q/n),q (mod) n}$, $dim vec X = n^2$ is the vector-reformulated unknown Matrix and $M$ a constant Matrix of coefficients that are obtained by similar procedure making a rank 4 Matrix to a rank 2 Matrix. After this, I can use the Theorem that if $det M = 0$ then nontrivial Solutions exists. Is this correct?
Are there other ways to determine whether a nontrivial solution exist for this system?
linear-algebra systems-of-equations
asked Jan 17 at 12:31
kryomaximkryomaxim
2,3621825
$endgroup$
add a comment |
$begingroup$
Given the ordinary matrix $X_{ij}$ of Dimension $n$ and some constant Matrices $A_{ij},B_{ij}$, find a solution of the System of equations
$X_{ij} = sum_{k,l=1}^n A_{ik}X_{kl}B_{lj}$
for all rows $i in { 1,dots,n }$ and columns $j in { 1,dots,n }$ that is different from Zero. Rewriting this equation as
$sum_{k,l=1}^n(delta_{ik}delta_{jl}-A_{ik}B_{lj})X_{kl} = 0$
and supposing that all Matrix entries can be expressed as a Vector of Dimension $n^2$, I can write
$Mvec X = 0$
where $(vec{X})_q = X_{ceiling(q/n),q (mod) n}$, $dim vec X = n^2$ is the vector-reformulated unknown Matrix and $M$ a constant Matrix of coefficients that are obtained by similar procedure making a rank 4 Matrix to a rank 2 Matrix. After this, I can use the Theorem that if $det M = 0$ then nontrivial Solutions exists. Is this correct?
Are there other ways to determine whether a nontrivial solution exist for this system?
linear-algebra systems-of-equations
asked Jan 17 at 12:31
kryomaximkryomaxim
2,3621825
$endgroup$
Given the ordinary matrix $X_{ij}$ of Dimension $n$ and some constant Matrices $A_{ij},B_{ij}$, find a solution of the System of equations
$X_{ij} = sum_{k,l=1}^n A_{ik}X_{kl}B_{lj}$
for all rows $i in { 1,dots,n }$ and columns $j in { 1,dots,n }$ that is different from Zero. Rewriting this equation as
$sum_{k,l=1}^n(delta_{ik}delta_{jl}-A_{ik}B_{lj})X_{kl} = 0$
and supposing that all Matrix entries can be expressed as a Vector of Dimension $n^2$, I can write
$Mvec X = 0$
where $(vec{X})_q = X_{ceiling(q/n),q (mod) n}$, $dim vec X = n^2$ is the vector-reformulated unknown Matrix and $M$ a constant Matrix of coefficients that are obtained by similar procedure making a rank 4 Matrix to a rank 2 Matrix. After this, I can use the Theorem that if $det M = 0$ then nontrivial Solutions exists. Is this correct?
Are there other ways to determine whether a nontrivial solution exist for this system?
linear-algebra systems-of-equations
linear-algebra systems-of-equations
asked Jan 17 at 12:31
kryomaximkryomaxim
2,3621825
asked Jan 17 at 12:31
kryomaximkryomaxim
2,3621825
asked Jan 17 at 12:31
kryomaximkryomaxim
2,3621825
asked Jan 17 at 12:31
kryomaximkryomaxim
2,3621825
asked Jan 17 at 12:31
kryomaximkryomaxim
2,3621825
2,3621825
add a comment |
add a comment |
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