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Many computational science and engineering problems result in complex-valued systems of linear equations. At the same time, a large portion of equation solver software is written to solve only real-valued systems. Casting complex systems into equivalent real formulations (ERFs) enables the use of real solvers and preconditioners. Each ERF has unique properties that make it desirable in certain instances. We created a mathematical framework that permits easy conversions between different ERFs. This will allow, for instance, one ERF to be used as a preconditioner and another ERF to be used to iteratively solve the linear system by simply switching back and forth between the forms through scaling and permuting. Such transformations between ERF forms are attractive for simultaneously exploiting spectral and symmetry properties in different phases of a solver. My thesis describes the specific diagonal and permutation matrices needed as well as how to transform from one ERF to another.