@Markiyan Hirnyk

hi there! I have a question about this please. Rather than using a specific 4x4 matrix T (as I did originally above),

I want to use a general 4x4 matrix for T. Then I want to solve S.R=~T subject to the given constraints

of eq1 and eq2 as above. I want to find what S and R must be in order to have a solution. I tried doing

this (and had the program terminate after one solution was found since I only need one, I do not need

them

all) and I got a solution. But the solution doesnt make sense since the unknown variables (the variables of

the matrices S and R) don't seem to be resolved. For example, the solution j = .... is in terms of entries of

S and R and thus doesnt seem to be resolved. I converted it to LaTeX:

Solving the equation $T = S \cdot R$ is equivalent to solving the

polynomial system

\begin{align*}

q & = \sum_{i = 1}^4 a_i s_i, \\

w & = \sum_{i = 1}^4 a_i t_i, \\

y & = \sum_{i = 1}^4 a_i r_i, \\

u & = \sum_{i = 1}^4 a_i l_i, \\

&\\

p & = \sum_{i = 1}^4 b_i s_i, \\

f & = \sum_{i = 1}^4 b_i t_i, \\

g & = \sum_{i = 1}^4 b_i r_i, \\

h & = \sum_{i = 1}^4 b_i l_i, \\

&\\

j & = \sum_{i = 1}^4 c_i s_i, \\

k & = \sum_{i = 1}^4 c_i t_i, \\

z & = \sum_{i = 1}^4 c_i r_i, \\

x & = \sum_{i = 1}^4 c_i l_i, \\

&\\

v & = \sum_{i = 1}^4 d_i s_i, \\

n & = \sum_{i = 1}^4 d_i t_i, \\

m & = \sum_{i = 1}^4 d_i r_i, \\

o & = \sum_{i = 1}^4 d_i l_i,

\end{align*}

subject to the constraints $a_i d_i = b_i c_i$ and $s_i l_i = t_i r_i$ for $i = 1, 2, 3, 4$.

Using Maple, we determined one solution (we had the program terminate once one

solution was determined) to be:

\[

{\it a_1}={\it a_1},

\]

\[

{\it a_2}={\it a_2},

\]

\[

{\it a_3}={\it

a_3},

\]

\[

{\it a_4}={\frac {{\it b_4}\,{\it c_4}}{{\it d_4}}},

\]

\[

{\it

b_1}=0,

\]

\[

{\it b_2}=0,

\]

\[

{\it b_3}=0,

\]

\[

{\it b_4}={\it b_4},

\]

\[

{\it c_1}={

\it c_1},

\]

\[

{\it c_2}={\it c_2},

\]

\[

{\it c_3}={\it c_3},

\]

\[

{\it c_4}={\it

c_4},

\]

\[

{\it d_1}=0,

\]

\[

{\it d_2}=0,

\]

\[

{\it d_3}=0,

\]

\[

{\it d_4}={\it d_4},

\]

\[

f={

\it b_4}\,{\it t_4},

\]

\[

g={\it b_4}\,{\it r_4},

\]

\[

h={\frac {{\it b_4}\,{

\it t_4}\,{\it r_4}}{{\it s_4}}},

\]

\[

j={\it c_1}\,{\it s_1}+{\it c_2

}\,{\it s_2}+{\it c_3}\,{\it s_3}+{\it c_4}\,{\it s_4},

\]

\[

k={\it

c_1}\,{\it t_1}+{\it c_2}\,{\it t_2}+{\it c_3}\,{\it t_3}+{\it

c_4}\,{\it t_4},

\]

\[

{\it l_1}={\frac {{\it t_1}\,{\it r_1}}{{\it s_1

}}},

\]

\[

{\it l_2}={\frac {{\it t_2}\,{\it r_2}}{{\it s_2}}},

\]

\[

{\it l_3}

={\frac {{\it t_3}\,{\it r_3}}{{\it s_3}}},

\]

\[

{\it l_4}={\frac {{\it

t_4}\,{\it r_4}}{{\it s_4}}},

\]

\[

m={\it d_4}\,{\it r_4},

\]

\[

n={\it d_4}

\,{\it t_4},

\]

\[

o={\frac {{\it d_4}\,{\it t_4}\,{\it r_4}}{{\it s_4}}

},

\]

\[

p={\it b_4}\,{\it s_4},

\]

\[

q={\frac {{\it a_1}\,{\it s_1}\,{\it d_4

}+{\it a_2}\,{\it s_2}\,{\it d_4}+{\it a_3}\,{\it s_3}\,{\it d_4

}+{\it b_4}\,{\it c_4}\,{\it s_4}}{{\it d_4}}},

\]

\[

{\it r_1}={\it

r_1},

\]

\[

{\it r_2}={\it r_2},

\]

\[

{\it r_3}={\it r_3},

\]

\[

{\it r_4}={\it r_4

},

\]

\[

{\it s_1}={\it s_1},

\]

\[

{\it s_2}={\it s_2},

\]

\[

{\it s_3}={\it s_3},

\]

\[

{

\it s_4}={\it s_4},

\]

\[{\it t_1}={\it t_1},

\]

\[

{\it t_2}={\it t_2},

\]

\[

{\it

t_3}={\it t_3},

\]

\[

{\it t_4}={\it t_4},

\]

\[

u={\frac {{\it a_1}\,{\it t_1

}\,{\it r_1}\,{\it s_2}\,{\it s_3}\,{\it d_4}\,{\it s_4}+{\it

a_2}\,{\it t_2}\,{\it r_2}\,{\it s_1}\,{\it s_3}\,{\it d_4}\,{

\it s_4}+{\it a_3}\,{\it t_3}\,{\it r_3}\,{\it s_1}\,{\it s_2}\,

{\it d_4}\,{\it s_4}+{\it b_4}\,{\it c_4}\,{\it t_4}\,{\it r_4}

\,{\it s_1}\,{\it s_2}\,{\it s_3}}{{\it s_1}\,{\it s_2}\,{\it

s_3}\,{\it d_4}\,{\it s_4}}},

\]

\[

v={\it d_4}\,{\it s_4},

\]

\[

w={\frac {{

\it a_1}\,{\it t_1}\,{\it d_4}+{\it a_2}\,{\it t_2}\,{\it d_4}+{

\it a_3}\,{\it t_3}\,{\it d_4}+{\it b_4}\,{\it c_4}\,{\it t_4}}{

{\it d_4}}},

\]

\[

x={\frac {{\it c_1}\,{\it t_1}\,{\it r_1}\,{\it s_2}

\,{\it s_3}\,{\it s_4}+{\it c_2}\,{\it t_2}\,{\it r_2}\,{\it s_1

}\,{\it s_3}\,{\it s_4}+{\it c_3}\,{\it t_3}\,{\it r_3}\,{\it

s_1}\,{\it s_2}\,{\it s_4}+{\it c_4}\,{\it t_4}\,{\it r_4}\,{

\it s_1}\,{\it s_2}\,{\it s_3}}{{\it s_1}\,{\it s_2}\,{\it s_3}

\,{\it s_4}}},

\]

\[

y={\frac {{\it a_1}\,{\it r_1}\,{\it d_4}+{\it a_2}

\,{\it r_2}\,{\it d_4}+{\it a_3}\,{\it r_3}\,{\it d_4}+{\it b_4}

\,{\it c_4}\,{\it r_4}}{{\it d_4}}},

\]

\[

z={\it c_1}\,{\it r_1}+{\it

c_2}\,{\it r_2}+{\it c_3}\,{\it r_3}+{\it c_4}\,{\it r_4}

\]