This paper presents a rank minimization heuristic for solving problems involving minimizing the rank of a matrix subject to linear matrix inequality (LMI) constraints. The heuristic replaces the nonconvex rank objective with the sum of the singular values of the matrix, known as the nuclear norm. This approach transforms the problem into a convex optimization problem, specifically a semidefinite program (SDP), which can be efficiently solved. The heuristic is motivated by the fact that the dual spectral norm is the convex envelope of the rank function on the set of matrices with norm less than one. This allows the heuristic to be interpreted as a relaxation of the original rank minimization problem. The paper demonstrates the effectiveness of this heuristic in the context of minimum order system approximation, where the goal is to find a lower-order system that approximates a given system within a specified error tolerance. The method is applied to the problem of minimum order system approximation, where the objective is to minimize the MacMillan degree (the sum of the ranks of the residue matrices) while satisfying frequency response constraints. The heuristic is extended to the complex case using Hermitian conjugates. The problem is then reformulated as an SDP, which can be solved using existing SDP solvers. The paper also provides a numerical example, showing that the heuristic can effectively reduce the order of a system while maintaining a desired level of accuracy. The results demonstrate that the dual spectral norm heuristic provides a lower bound on the optimal rank objective, offering a practical solution to the challenging problem of rank minimization.This paper presents a rank minimization heuristic for solving problems involving minimizing the rank of a matrix subject to linear matrix inequality (LMI) constraints. The heuristic replaces the nonconvex rank objective with the sum of the singular values of the matrix, known as the nuclear norm. This approach transforms the problem into a convex optimization problem, specifically a semidefinite program (SDP), which can be efficiently solved. The heuristic is motivated by the fact that the dual spectral norm is the convex envelope of the rank function on the set of matrices with norm less than one. This allows the heuristic to be interpreted as a relaxation of the original rank minimization problem. The paper demonstrates the effectiveness of this heuristic in the context of minimum order system approximation, where the goal is to find a lower-order system that approximates a given system within a specified error tolerance. The method is applied to the problem of minimum order system approximation, where the objective is to minimize the MacMillan degree (the sum of the ranks of the residue matrices) while satisfying frequency response constraints. The heuristic is extended to the complex case using Hermitian conjugates. The problem is then reformulated as an SDP, which can be solved using existing SDP solvers. The paper also provides a numerical example, showing that the heuristic can effectively reduce the order of a system while maintaining a desired level of accuracy. The results demonstrate that the dual spectral norm heuristic provides a lower bound on the optimal rank objective, offering a practical solution to the challenging problem of rank minimization.