create a loop then solve system of equations

I will explain what i want then i will show my code which is not work as what i want

I have a matrix A given,

n=size(A,1) %l let say n=3

esu=triu(A,1)

[row,col,v]=find(esu) % to spicify only the nonzeros element in the strictly upper diagonal

m=size(v,1)

and I want to define new n×n matrix E_ij such that all elements in E_ij equal zeros except at the position (i,j) and the position(j,i) equal to 1/sqrt(2)

Then I want to apply that each value of v at the position k is equal to trace (E_ij* L(X) )where L(X) lets call it M is n×n matrix with unknown elements

note : X is n-1×n-1 unknown matrix and this is my goal to find it

at the end I want to solve these equations togethor

so for example :

if v=[3 ;2]

so i will have two equations:

v(1)= trace (E_12*M)

thus , 3=trace([0 1/sqrt(2) 0; 1/sqrt(2) 0 0; 0 0 0]*[m11 m12 m13;m21 m22 m23; m31 m32 m33])

similarly

v(2)= trace (E_13*M)

thus 2=trace([0 0 1/sqrt(2) ; 0 0 0; 1/sqrt(2) 0 0]*[m11 m12 m13;m21 m22 m23; m31 m32 m33])

A=[0 1 0;1 0 2; 0 2 0]
n= size(A,1)
E = zeros(n,n);
E(i,j) = 1/sqrt(2);
E(j,i) = 1/sqrt(2);
esu=triu(A,1);
[row,col,v]=find(esu);  % corresp. subscripts
m=size(v,1);
A= zeros(m, n^2)
for k = 1:m
    [i,j] = deal(row(k), col(k))
    E = zeros(n,n)
    E(i,j) = 1/sqrt(2)
    E(j,i) = 1/sqrt(2)
    M = trace(E*L(X))-v(k)==0
    M*x=b
    M(k,:) = reshape(E, 1, [])
end
b = v
x = pinv(M) * b
X = reshape(x, n, n)

Answer 1

[Shushant]

Hi @Hajar Alshaikh,

According to my understanding, you want to find an Unknown Matrix "X" of size "nxn" using a set of equations which are stored in "M". To accomplish this, I suggest that you use "syms" to solve the system of equations.

Here is the documentation for "syms" and documentation for solving system of equations.

Create symbolic scalar variables and functions, and matrix variables and functions - MATLAB syms (mathworks.com)

Solve System of Linear Equations - MATLAB & Simulink (mathworks.com)

Below is a sample code which might give you some idea on how to use syms and solve the system of equations .

A=[0 1 0;1 0 2; 0 2 0];
n= size(A,1);
syms X [n,n];
esu=triu(A,1);
[row,col,v]=find(esu);  % corresp. subscripts
m=size(v,1);
A = zeros(m, n^2);
for k = 1:m
    [i,j] = deal(row(k), col(k));
    E = zeros(n,n);
    E(i,j) = 1/sqrt(2);
    E(j,i) = 1/sqrt(2);
    M(k) = trace(E*X)==v(k);
end
sol = solve([M(1:k)], X);
x = struct2cell(sol);
x = double(vertcat(x{:}));
x = reshape(x, n, n)

x = 3×3

0 0 0 1.4142 0 0 0 2.8284 0