Please can you help me?
I have such a problem: (next text written in tex)
There are 4 variables $X_1$-$X_4$, they are part of an polynomial. The coeffiecients of these polynomials $a-d$ are
different for each line of the matrix (thus they have different numbers $1-4$, $a_1\ne a_4$ for instance).
This matrix has an subscript layer$_i$ which means that this matrix is written for an layer number $i$ and thus $X_1$ in layer 1
is not the same like $X_1$ in layer N (that means $X_1$ in the matrix on the left side is not the same like $X_1$ on the rigth side of the eq.).
I know each coeffiecient $a-d$ in each matrix $1-N$ for $1-N$ layers. The symbols $M_i$, $N_i$ and $L_i$
describe another 4 x 4 ($M_i, L_i$) or 1x4 ($N_i$) matrices which are unique for given layer $i$.
\begin{equation}
\begin{split}&
\left( \begin{array}{c}
a_1X_1+b_2X_1+c_1X_3+d_1X_4 \\
a_2X_1+b_2X_1+c_2X_3+d_2X_4 \\
a_3X_1+b_3X_1+c_3X_3+d_3X_4 \\
a_4X_1+b_4X_1+c_4X_3+d_4X_4 \end{array} \right)_{layer_{N}}=\\
&=M_{N-1}\cdot\left(...\,M_1\cdot\left(L_1\cdot\left(\begin{array}{c}
a_1X_1+b_2X_1+c_1X_3+d_1X_4 \\
a_2X_1+b_2X_1+c_2X_3+d_2X_4 \\
a_3X_1+b_3X_1+c_3X_3+d_3X_4 \\
a_4X_1+b_4X_1+c_4X_3+d_4X_4 \end{array}\right)_{layer_1}+N_1\right)...+N_{N-1}\right)\\
\end{split}
\end{equation}
So as you can see, the $X_1-X_4$ in layer 1 and $X_1-X_4$ in layer N are my 8 unknowns I need to
find. These are 8 variables for 4 eq. actually. But I have additional conditions for $X_1-X_4$ in 1st layer and $X_1-X4$ in Nth
layer (4 additional eq.), so I have 8 eq. for 8 unknowns. I would write it as a additional 4 lines in my matrix schema
in order to solve the system.
My problem is, however, how to solve such system of eq. (1): the unknowns
in the 1st layer on the right side (matrix $layer_1$) are hidden in many matrix multiplications. The matrices $M_i,N_i,L_i$ are filled with known coeffiecient (numbers)
but the central matrix is filled by mixture of numbers and unknowns... Any idea? I know how to solve for instance the left side
of the eq. because its easy (e.g. Gauss eliminiation or so) but what to do with the right side (and it's clear, it must be
finally on the left side and everything what does not contain $X_1-X_4$ on the right side and the use something like Gauss elimination..)
Many thanks for help - for any idea!
I have such a problem: (next text written in tex)
There are 4 variables $X_1$-$X_4$, they are part of an polynomial. The coeffiecients of these polynomials $a-d$ are
different for each line of the matrix (thus they have different numbers $1-4$, $a_1\ne a_4$ for instance).
This matrix has an subscript layer$_i$ which means that this matrix is written for an layer number $i$ and thus $X_1$ in layer 1
is not the same like $X_1$ in layer N (that means $X_1$ in the matrix on the left side is not the same like $X_1$ on the rigth side of the eq.).
I know each coeffiecient $a-d$ in each matrix $1-N$ for $1-N$ layers. The symbols $M_i$, $N_i$ and $L_i$
describe another 4 x 4 ($M_i, L_i$) or 1x4 ($N_i$) matrices which are unique for given layer $i$.
\begin{equation}
\begin{split}&
\left( \begin{array}{c}
a_1X_1+b_2X_1+c_1X_3+d_1X_4 \\
a_2X_1+b_2X_1+c_2X_3+d_2X_4 \\
a_3X_1+b_3X_1+c_3X_3+d_3X_4 \\
a_4X_1+b_4X_1+c_4X_3+d_4X_4 \end{array} \right)_{layer_{N}}=\\
&=M_{N-1}\cdot\left(...\,M_1\cdot\left(L_1\cdot\left(\begin{array}{c}
a_1X_1+b_2X_1+c_1X_3+d_1X_4 \\
a_2X_1+b_2X_1+c_2X_3+d_2X_4 \\
a_3X_1+b_3X_1+c_3X_3+d_3X_4 \\
a_4X_1+b_4X_1+c_4X_3+d_4X_4 \end{array}\right)_{layer_1}+N_1\right)...+N_{N-1}\right)\\
\end{split}
\end{equation}
So as you can see, the $X_1-X_4$ in layer 1 and $X_1-X_4$ in layer N are my 8 unknowns I need to
find. These are 8 variables for 4 eq. actually. But I have additional conditions for $X_1-X_4$ in 1st layer and $X_1-X4$ in Nth
layer (4 additional eq.), so I have 8 eq. for 8 unknowns. I would write it as a additional 4 lines in my matrix schema
in order to solve the system.
My problem is, however, how to solve such system of eq. (1): the unknowns
in the 1st layer on the right side (matrix $layer_1$) are hidden in many matrix multiplications. The matrices $M_i,N_i,L_i$ are filled with known coeffiecient (numbers)
but the central matrix is filled by mixture of numbers and unknowns... Any idea? I know how to solve for instance the left side
of the eq. because its easy (e.g. Gauss eliminiation or so) but what to do with the right side (and it's clear, it must be
finally on the left side and everything what does not contain $X_1-X_4$ on the right side and the use something like Gauss elimination..)
Many thanks for help - for any idea!