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I have an issue of how to interpret the solution of the following nonlinear system in a matrix:

$F=PX^{-1}AX+wL$

where $w>0$ is a scalar, $F=(F_1,F_2)$, $P=(P_1,P_2)$, and $L=(L_1,L_2)$ are all a row vector and $A$ is a $2\times2$ matrix of $a_{ij}$ and $X=diag(x_1,x_2)$ is a diagonal matrix. All the matrices and vectors are strictly positive.

$X$ is the unknown while the rest are all taken as given. To find the solution for $x_1$ and $x_2$, I have the following Mathematica code:

A = {{a11, a12}, {a21, a22}};
l = {{l1, l2}};
F = {{F1, F2}};
P = {{P1, P2}};
X = DiagonalMatrix[{x1, x2}];
Solve[F == P.Inverse[X].A.X + w*l, {x1, x2}]

The result is simply:

{}

Does this mean that the above equation is always true regardless of the value of $x_1$ and $x_2$? But we can verify that this is not the case. To see this, suppose $X$ given and $P$ unknown; the code to find the solution for $P$ is

Solve[F == P.Inverse[X].A.X + w*l, {P1, P2}]

which yields

{{P1 -> -((-a21 F2 x1 + a21 l2 w x1 + a22 F1 x2 - a22 l1 w x2)/((a12 a21 - a11 a22) x2)), P2 -> (-a11 F2 x1 + a11 l2 w x1 + a12 F1 x2 - a12 l1 w x2)/((a12 a21 - a11 a22) x1)}}

If you look at this solution for $P_1$ and $P_2$, it does not seem to be the case that the values of $x_1$ and $x_2$ do not matter; i.e. changing the values of $x_1$ and $x_2$ would change the solution for $P_1$ and $P_2$. This observation seems to contradict the solution when $X$ was unknown.

Am I missing something?

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  • $\begingroup$ Replacing Solve with Reduce doesn't give an empty list as a solution, but getting the output from Reduce into a matrix form can be challenging. Mathematica has internal default rules to interpret a list as a row or a column and adding extra {} does not change this. $\endgroup$ – Bill Jun 24 at 19:44
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Look at the equations you're trying to solve:

F == P.Inverse[X].A.X + w l // Simplify

{{F1, F2}} == {{a11 P1 + l1 w + (a21 P2 x1)/x2, a22 P2 + l2 w + (a12 P1 x2)/x1}}

So, from:

F1 == a11 P1 + l1 w + a21 P2 (x1/x2)

you can solve for x1/x2:

x1x2sol = First @ Solve[F1 == a11 P1 + l1 w + a21 P2 (x1/x2) /. x1/x2->x1x2, x1x2]

{{x1x2 -> (F1 - a11 P1 - l1 w)/(a21 P2)}}

Plugging this into the second equation:

F2 == a22 P2 + l2 w + a12 P1/x1x2 /. x1x2sol

{F2 == a22 P2 + l2 w + (a12 a21 P1 P2)/(F1 - a11 P1 - l1 w)}

There is no way to solve this equation for x1 or x2. Hence, there is no solution for your equation. This is what {} means, as you can see from the documentation:

enter image description here

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  • $\begingroup$ Thanks, Carl! Follow-up questions: (1) Can we know why the system has no solution? Any implication? (2) Can we generalize this result and say that there is no solution in the $n \times n$ case, either? $\endgroup$ – ppp Jun 25 at 2:57

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