Let $A$ be an $n \times n$ positive semi-definite matrix with eigenvalues $\lambda_1,\ldots,\lambda_n$ and corresponding eigenvectors $v_1,\ldots,v_n$. Suppose that $\lambda_1 > \lambda_2 > \cdots > \lambda_n$ and that the eigenvectors have unit 2-norms. Let $c$ be a strictly positive real number. Then, the minimum value of the function $f(X) = \operatorname{trace}(AX)$ over all $n \times n$ Hermitian matrices $X$ satisfying $\|X\|_* \le c$ is attained at $X =$ ___. Here, $\|X\|_*$ denotes the nuclear norm of the matrix $X$, i.e., the sum of the singular values of $X$. Express your answer in terms of $\lambda_i$, $v_i$, $v_i^*$, $c$, and numbers, where $v_i^*$ denotes the conjugate transpose of $v_i$.