  Random matrix theory allows one to deduce the eigenvalue spectrum of a large
matrix given only statistical information about its elements. Such results
provide insight into what factors contribute to the stability of complex
dynamical systems. In this letter, we study the eigenvalue spectrum of an
ensemble of random matrices with correlations between any pair of elements. To
this end, we introduce an analytical method that maps the resolvent of the
random matrix onto the response functions of a linear dynamical system. The
response functions are then evaluated using a path integral formalism, enabling
us to make deductions about the eigenvalue spectrum. Our central result is a
simple, closed-form expression for the leading eigenvalue of a large random
matrix with generalised correlations. This formula demonstrates that
correlations between matrix elements that are not diagonally opposite, which
are often neglected, can have a significant impact on stability.
