# Symbolically solve system of equations involving matrix

I need to solve the following system of equations symbolically for a_i, b_i, and c_i:

$$\beta_1 (1-a_i-b_i-c_i) \sum_{j=1}^{n}A_{i,j}(a_j+c_j)-\delta_1 a_i+\delta_2 c_i - \varepsilon \beta_2 a_i \sum_{j=1}^{n}B_{i,j}(b_j+c_j)=0,\\ \beta_2 (1-a_i-b_i-c_i) \sum_{j=1}^{n}B_{i,j}(b_j+c_j)-\delta_2 b_i+\delta_1 c_i - \varepsilon \beta_1 b_i \sum_{j=1}^{n}A_{i,j}(a_j+c_j)=0,\\ \varepsilon \beta_2 a_i \sum_{j=1}^{n}B_{i,j}(b_j+c_j) + \varepsilon \beta_1 b_i \sum_{j=1}^{n}A_{i,j}(a_j+c_j)-(\delta_1+\delta_2)c_i=0.$$

Here, $a_i$, $b_i$ and $c_i$ are considered to be probabilities for some node $i$ while $A$ and $B$ are adjacency matrices such that $A_{i,j}$ denotes whether $i$ is connected to $j$ or not. I have attempted to solve the above equations using the following code, but failed:

Remove["Global*"]
Solve[{Subscript[β,
1] (1 - Subscript[a, i] - Subscript[b, i] - Subscript[c, i]) Sum[
Subscript[A, i, j] (Subscript[a, j] + Subscript[c, j]), {j, 1,
n}] - Subscript[δ, 1] Subscript[a, i] +
Subscript[δ, 2] Subscript[c,
i] - ϵ Subscript[β, 2] Subscript[a, i]
Sum[Subscript[B, i, j] (Subscript[b, j] + Subscript[c, j]), {j,
1, n}] == 0,
Subscript[β,
2] (1 - Subscript[a, i] - Subscript[b, i] - Subscript[c, i]) Sum[
Subscript[B, i, j] (Subscript[b, j] + Subscript[c, j]), {j, 1,
n}] - Subscript[δ, 2] Subscript[b, i] +
Subscript[δ, 1] Subscript[c,
i] - ϵ Subscript[β, 1] Subscript[b, i]
Sum[Subscript[A, i, j] (Subscript[a, j] + Subscript[c, j]), {j,
1, n}] == 0,
ϵ Subscript[β, 2] Subscript[a, i]
Sum[Subscript[B, i, j] (Subscript[b, j] + Subscript[c, j]), {j,
1, n}] + ϵ Subscript[β, 1] Subscript[b, i]
Sum[Subscript[A, i, j] (Subscript[a, j] + Subscript[c, j]), {j,
1, n}] - (Subscript[δ, 1] + Subscript[δ,
2]) Subscript[c, i] == 0,}, {Subscript[a, i], Subscript[b, i], Subscript[c, i]}]

Could anyone kindly tell me how to obtain the symbolic solutions for this system. Thank you in advance!

• Make up yer mind, are these differential or difference equations? For the former, you use DSolve[]; for the latter, RSolve[]. Commented May 15, 2015 at 8:19
• These are differential equations where the $t$ variable has been dropped as these equations are investigated in steady-state condition. So, using DSolve does not help. Commented May 15, 2015 at 8:23
• It is a algebraic system which includes Schur product.I think you could send it to Mathematics Stackexchange.:) Commented May 15, 2015 at 10:30

Not an answer but too long for comment:

If I modify slightly your input and choose 'n=2

n = 2;
Solve[Table[{Subscript[β,
1] (1 - Subscript[a, i] - Subscript[b, i] -
Subscript[c, i]) Sum[
Subscript[A, i, j] (Subscript[a, j] + Subscript[c, j]), {j, 1,
n}] - Subscript[δ, 1] Subscript[a, i] +
Subscript[δ, 2] Subscript[c,
i] - ϵ Subscript[β, 2] Subscript[a, i] Sum[
Subscript[B, i, j] (Subscript[b, j] + Subscript[c, j]), {j, 1,
n}] == 0,
Subscript[β,
2] (1 - Subscript[a, i] - Subscript[b, i] -
Subscript[c, i]) Sum[
Subscript[B, i, j] (Subscript[b, j] + Subscript[c, j]), {j, 1,
n}] - Subscript[δ, 2] Subscript[b, i] +
Subscript[δ, 1] Subscript[c,
i] - ϵ Subscript[β, 1] Subscript[b, i] Sum[
Subscript[A, i, j] (Subscript[a, j] + Subscript[c, j]), {j, 1,
n}] == 0, ϵ Subscript[β, 2] Subscript[a,
i] Sum[Subscript[B, i,
j] (Subscript[b, j] + Subscript[c, j]), {j, 1,
n}] + ϵ Subscript[β, 1] Subscript[b, i] Sum[
Subscript[A, i, j] (Subscript[a, j] + Subscript[c, j]), {j, 1,
n}] - (Subscript[δ, 1] +
Subscript[δ, 2]) Subscript[c, i] == 0}, {i, n}] //
Flatten, Table[{Subscript[a, i], Subscript[b, i],
Subscript[c, i]}, {i, n}] // Flatten]

I see that even in this case we have 6 quadratic equations

in the variable

so it is unlikely there is an analytical solution of the general case.