LMI appears unsymmetric

1

I am trying to minimize the position error bound (PEB) with power constraints. The code is below:

omega_g = exp(1i*2*pi.*rand(N_ris,M));

cvx_solver sedumi
cvx_begin sdp
variable X(N_t, N_t) hermitian

expressions J_gamma(2,2) PEB(2,2)

for m=1:M
 k1=2/(sigma^2);
 qwmn=abs((betaR+1j*betaI)*x_m(m)*alpha)^2;
 k3=abs(x_m(m)*alpha)^2;
 k4=(betaR-1j*betaI)*k3;
 J_gamma(1,1) = J_gamma(1,1) + k1*qwmn*(a_K_a)'*a_K_a*cdot_ris'*omega_g(:,m)* 
  (omega_g(:,m))'*cdot_ris*a_B_d'*X*a_B_d; %(:,:,m)
 J_gamma(2,2) = J_gamma(2,2) + k1*qwmn*(adot_K_a')*adot_K_a*c_ris'*omega_g(:,m)* 
 (omega_g(:,m))'*c_ris*(a_B_d')*X*a_B_d;
 J_gamma(1,2) = J_gamma(1,2) + k1*qwmn*(a_K_a)'*adot_K_a*c_ris'*omega_g(:,m)* 
 (omega_g(:,m))'*cdot_ris*a_B_d'*X*a_B_d;
 J_gamma(2,1) = (J_gamma(1,2))';
end

 PEB = T*J_gamma*transpose(T);

% Objective: Minimize the Position Error Bound (PEB)
% PEB is based on the determinant of the FIM
minimize(trace_inv(PEB)) % Minimize the PEB

subject to   

 X == hermitian_semidefinite(N_t);    
real(trace(X)) <= 1;

cvx_end

I am getting a warning message as follows: "This linear matrix inequality appears to be unsymmetric. This is
very likely an error that will produce unexpected results. Please check
the LMI; and, if necessary, re-enter the model. "

Can someone please help me where I am getting the model wrong? I checked the PEB matrix and is symmetric.

2

I think the culprit is use of non-conjugate transpose, transpose(T), instead of conjugate transpose T' . If T is not real, that makes PEB unsymmetric (non-hermitian). So the PEV matrix is not symmetric.

T=rand(2,2)+i*rand(2,2);
cvx_begin sdp
variable X(2,2) hermitian;
% will generate unsymmetric warning
trace_inv(T*X*transpose(T))

Warning: This linear matrix inequality appears to be unsymmetric. This is
very likely an error that will produce unexpected results. Please check
the LMI; and, if necessary, re-enter the model.

In cvxprob/newcnstr (line 241)
In >= (line 21)
In cvx/trace_inv (line 28)

ans =
cvx convex expression (scalar)

% no warning
trace_inv(T*X*T')

ans =
cvx convex expression (scalar)

3

Or you can use ctranspose(T)

4

Thank you very much for the quick reply