# Solve a matricial equation, i.e a system of 16 equations with 16 unknown : Error when call solve function

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petit on 14 Jul 2021
I wonder under which conditions I could have equality between the 2 matrices 4x4.
I build my 2 matrixes from which I would like to get an equality. But to get this equality, I have to solve an unknow matrix 4x4 whose terms are present in the 2 matrices 4x4.
To show you a practical example, I put below a small Matlab script that tries to follow all the reasoning explained above :
clear;
clc;
% Big_1 Fisher :
FISH_Big_1_SYM = sym('sp_', [5,5], 'real');
% Force symmetry for Big_1
FISH_Big_1_SYM = tril(FISH_Big_1_SYM.') + triu(FISH_Big_1_SYM,1);
% Big_2 Fisher :
FISH_Big_2_SYM = sym('sp_', [5,5], 'real');
% Force symmetry for Big_2
FISH_Big_2_SYM = tril(FISH_Big_2_SYM.') + triu(FISH_Big_2_SYM,1);
% Jacobian 1
J_1_SYM = sym('j_', [5,5], 'real');
% Jacobian 2
J_2_SYM = sym('j_', [5,5], 'real');
% Remove 4th row/column
J_2_SYM(4,:) = [];
J_2_SYM(:,4) = [];
%%%%%%%% Method 1 : projection before marginalisation %%%%%%%%%
% Projection
FISH_proj_1 = J_1_SYM'*FISH_Big_1_SYM*J_1_SYM;
% Check size : 5x5
size(FISH_proj_1)
% Invert Fisher_2
COV_Big_1_SYM = inv(FISH_Big_1_SYM);
% Marginalisation
COV_Big_1_SYM(4,:) = [];
COV_Big_1_SYM(:,4) = [];
FISH_Big_1_SYM = inv(COV_Big_1_SYM);
%%%%%%%% Method 2 : projection after marginalisation %%%%%%%%%
% Invert Fisher_2
COV_Big_2_SYM = inv(FISH_Big_2_SYM);
% Remove 4th row/column
COV_Big_2_SYM(4,:) = [];
COV_Big_2_SYM(:,4) = [];
% Re-inverse
FISH_Big_2_SYM_new = inv(COV_Big_2_SYM);
% Projection 2x2
FISH_proj_2 = J_2_SYM'*FISH_Big_2_SYM_new*J_2_SYM;
% Check size : 4x4
size(FISH_proj_2)
% Remove 4th row/column of Fisher matrix method 1
FISH_proj_1(4,:) = [];
FISH_proj_1(:,4) = [];
% Test equality between 2 matrices
isequal(FISH_proj_1,FISH_proj_2);
% Matricial equation to solve
eqn = FISH_proj_1 == FISH_proj_2;
% Solving : Jacobian 4x4 unknown
[solx, parameters, conditions] = solve(eqn, J_2_SYM, 'ReturnConditions', true);
solx
Once I finsished to build my 2 matrices FISH_proj_1 and FISH_proj_2, I get the error in "solve" function :
Error using sym/solve (line 279)
Inconsistent output with 1 variables for input argument with 16 variables.
Error in demo_projection_dev (line 62)
[solx, parameters, conditions] = solve(eqn, J_2_SYM, 'ReturnConditions', true);
The unknown is a Matrix 4x4 named J_2_SYM.
How to debug this error with solve function ? Is the syntaw right if I put J_2_SYM directly in solve function call ?
Best regards

R2020a

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