How to make this problem convex and solve it using CVX?

Question

How to make this problem convex so that it can be solved by semidefinite programming in cvx? Given $A\in\mathbb{C}^{n\times n}$

\begin{array}{ll} \underset{X\in\mathbb{C^{n\times n}}}{\text{maximize}} & \frac{\mathrm{trace}\big((A^*A+I)X\big)}{\mathrm{trace}\big((A+A^*)X\big)}.\\ \text{subject to} & \mathrm{trace}\big((A-A^*)X\big)=0,\\&\mathrm{trace}(X)=1,\\&X\geq0.\end{array}

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I have tried to let $\mathrm{trace}\big((A+A^*)X\big)=1/k$, so that I can rewrite above problem as

\begin{array}{ll} \underset{X\in\mathbb{C^{n\times n}}, k\in\mathbb{R}}{\text{maximize}} & k\times\mathrm{trace}\big((A^*A+I)X\big).\\ \text{subject to} & k\times\mathrm{trace}\big((A+A^*)X\big)=1\\&\mathrm{trace}\big((A-A^*)X\big)=0,\\&\mathrm{trace}(X)=1,\\&X\geq0.\end{array}

CVX code from Matlab:

    n=length(A);
    A1=A'*A+eye(n,n);
    A2=0.5*(A+A'); 
    A3=0.5*(A-A'); 

    cvx_begin
    variable X(n,n) hermitian
    variable k 
    minimize(-k*trace(A1*X));
    subject to
    k*trace(A2*X)==1;
    trace(A3*X)==0;
    trace(X)==1;
    X == hermitian_semidefinite(n);

ERROR code:

Error using  .*  (line 262)
Disciplined convex programming error:
    Invalid quadratic form(s): not a square.


Error in  *  (line 36)
    z = feval( oper, x, y );

Error in QCQP_SDR_GM (line 11)
minimize(-k*trace(A1*X));