# How to find solutions for a matrix equality, i.e a system of equations ?

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petit on 10 Apr 2021
Edited: petit on 13 Apr 2021
I would like to make work a small script. This Matlab code is expected to solve a matricial equation, but by equality of 2 final matrixes, this is about solve 4 equations with 3 unknown (since I have initially an array of 2x2 unknown but this matrix of unknown is symmetric, so actually there are 3 unkonwns).
But for the moment, Matlab doesn't succeed in finding the solutions.
Here is the Matlab script :
clear;
clc;
format long;
% 2 Fisher Matrixes symbolic : FISH_GCsp_SYM, : 1 cosmo params + 1 bias spectro put for common
% FISH_XC_SYM : 1 cosmo params + 2 bias photo correlated
% Spectro
% First parameter : cosmo , second : bias spectro
FISH_GCsp_SYM = sym('sp1_', [4,4], 'real');
FISH_GCsp_SYM(3:4,3:4) = 0
FISH_GCsp_SYM(1:2,3:4) = 0
FISH_GCsp_SYM(3:4,1:2) = 0
% Force symmetry
FISH_GCsp_SYM = tril(FISH_GCsp_SYM.') + triu(FISH_GCsp_SYM,1)
% First parameter : cosmo , second : 2 bias photo correlated
FISH_XC_SYM = sym('xc1_', [3,3], 'real');
FISH_XC_SYM(1,2) = 0;
FISH_XC_SYM(2,1) = 0;
FISH_XC_SYM(2,2) = 0;
FISH_XC_SYM = tril(FISH_XC_SYM.') + triu(FISH_XC_SYM,1);
% FISH_ALL1 : brutal marginalization on Fisher matrix
FISH_ALL1(1:2,1:2) = FISH_GCsp_SYM(1:2,1:2) + FISH_XC_SYM(1:2:3);
% FISH_SYM2 : from big Fisher matrix
FISH_GCsp_SYM = sym('sp1_', [4,4], 'real');
FISH_GCsp_SYM(3:4,3:4) = 0;
FISH_GCsp_SYM(1:2,3:4) = 0;
FISH_GCsp_SYM(3:4,1:2) = 0;
% Force symmetry
FISH_GCsp_SYM = tril(FISH_GCsp_SYM.') + triu(FISH_GCsp_SYM,1);
% First parameter : cosmo , second : 2 bias photo correlated
FISH_XC_SYM = sym('xc1_', [4,4], 'real');
FISH_XC_SYM(1,2) = 0;
FISH_XC_SYM(2,1) = 0;
FISH_XC_SYM(2,2) = 0;
FISH_XC_SYM = tril(FISH_XC_SYM.') + triu(FISH_XC_SYM,1);
% Adding new observable "O" terms
FISH_O_SYM = sym('o_', [2,2], 'real');
FISH_O_SYM = tril(FISH_O_SYM.') + triu(FISH_O_SYM,1);
% Addition of second Fisher matrix
FISH_SYM2 = FISH_GCsp_SYM + FISH_XC_SYM;
FISH_SYM2(2,2) = FISH_SYM2(2,2) + FISH_O_SYM(1,1);
FISH_SYM2(3,3) = FISH_SYM2(3,3) + FISH_O_SYM(2,2);
FISH_SYM2(2,3) = FISH_SYM2(2,3) + FISH_O_SYM(1,2);
FISH_SYM2(3,2) = FISH_SYM2(3,2) + FISH_O_SYM(2,1);
% Marginalize FISH_SYM2 in order to get back a 2x2 matrix
% Invert to marginalyze
COV_ALL = inv(FISH_SYM2);
% Marginalize
COV_ALL([3,4],:) = [];
COV_ALL(:,[3,4]) = [];
FISH_ALL2 = inv(COV_ALL);
FISH_ALL1
FISH_ALL2
% Matricial equation to solve
eqn = FISH_ALL1 == FISH_ALL2;
% Solving : matrix1 equal to matrix2
sol = solve(eqn, [FISH_O_SYM(1,1),FISH_O_SYM(2,2),FISH_O_SYM(1,2)]);
You can try it and see there are no found solutions. The context is about Fisher formalism. From my first analysis, I get 2 matrixes which have to be equal. I introduce in the second the unkonws o_1_1, o_1_2 and o_2_2 : So finally, I have a system of 4 equations with 3 unknows to solve.
If someone could help me to fix it, I would be grateful.
petit on 12 Apr 2021
Sorry, I did a mistake in the initial script.
Now, I get the following system of equations with unknows o_1_1, o_1_2, o_2_1 and o_2_2 :
FISH_ALL1 =
[ sp1_1_1 + xc1_1_1, sp1_1_2 + xc1_1_3]
[ sp1_1_2 + xc1_1_1, sp1_2_2 + xc1_1_3]
FISH_ALL2 =
[ -(o_2_2*xc1_1_4^2 + sp1_1_1*xc1_3_4^2 + xc1_1_1*xc1_3_4^2 + xc1_1_4^2*xc1_3_3 + xc1_1_3^2*xc1_4_4 - o_2_2*sp1_1_1*xc1_4_4 - o_2_2*xc1_1_1*xc1_4_4 - sp1_1_1*xc1_3_3*xc1_4_4 - 2*xc1_1_3*xc1_1_4*xc1_3_4 - xc1_1_1*xc1_3_3*xc1_4_4)/(- xc1_3_4^2 + o_2_2*xc1_4_4 + xc1_3_3*xc1_4_4), -(sp1_1_2*xc1_3_4^2 - o_2_2*sp1_1_2*xc1_4_4 - o_1_2*xc1_1_4*xc1_3_4 + o_2_2*xc1_1_4*xc1_2_4 + o_1_2*xc1_1_3*xc1_4_4 - sp1_1_2*xc1_3_3*xc1_4_4 - xc1_1_3*xc1_2_4*xc1_3_4 - xc1_1_4*xc1_2_3*xc1_3_4 + xc1_1_4*xc1_2_4*xc1_3_3 + xc1_1_3*xc1_2_3*xc1_4_4)/(- xc1_3_4^2 + o_2_2*xc1_4_4 + xc1_3_3*xc1_4_4)]
[ -(sp1_1_2*xc1_3_4^2 - o_2_2*sp1_1_2*xc1_4_4 - o_1_2*xc1_1_4*xc1_3_4 + o_2_2*xc1_1_4*xc1_2_4 + o_1_2*xc1_1_3*xc1_4_4 - sp1_1_2*xc1_3_3*xc1_4_4 - xc1_1_3*xc1_2_4*xc1_3_4 - xc1_1_4*xc1_2_3*xc1_3_4 + xc1_1_4*xc1_2_4*xc1_3_3 + xc1_1_3*xc1_2_3*xc1_4_4)/(- xc1_3_4^2 + o_2_2*xc1_4_4 + xc1_3_3*xc1_4_4), -(o_1_1*xc1_3_4^2 + o_2_2*xc1_2_4^2 + o_1_2^2*xc1_4_4 + sp1_2_2*xc1_3_4^2 + xc1_2_4^2*xc1_3_3 + xc1_2_3^2*xc1_4_4 - o_1_1*o_2_2*xc1_4_4 - o_2_2*sp1_2_2*xc1_4_4 - 2*o_1_2*xc1_2_4*xc1_3_4 + 2*o_1_2*xc1_2_3*xc1_4_4 - o_1_1*xc1_3_3*xc1_4_4 - sp1_2_2*xc1_3_3*xc1_4_4 - 2*xc1_2_3*xc1_2_4*xc1_3_4)/(- xc1_3_4^2 + o_2_2*xc1_4_4 + xc1_3_3*xc1_4_4)]

R2020a

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