A major advance in linear systems theory over the last decade has been a formalism for converting systems problems to matrix inequalities. In this tutorial paper we describe computer algebra algorithms, methodology, and implementation which allows users to convert many systems problems to linear matrix inequalities (LMIs). We shall focus on computer algebra methodology which can assist the user in producing LMIs for control design. We provide a step-by-step computer derivation of LMI formulas for the design of linear time-invariant dynamic controllers that achieve a prespecified performance measured by the H-infinity norm of a certain closed loop transfer function.