  Linear matrix inequalities (LMIs) commonly appear in systems, stability, and
control applications. Many analysis and synthesis problems in these areas can
be solved as feasibility or optimization problems subject to LMI constraints.
Although most well-known LMI properties and manipulation tricks, such as the
Schur complement and the congruence transformation, can be found in standard
references, many useful LMI properties are scattered throughout the literature.
The purpose of this document is to collect and organize properties, tricks, and
applications related to LMIs from a number of references together in a single
document. In this sense, the document can be thought of as an "LMI
encyclopedia" or "LMI cookbook." Proofs of the properties presented in this
document are not included when they can be found in the cited references in the
interest of brevity. Illustrative examples are included whenever necessary to
fully explain a certain property. Multiple equivalent forms of LMIs are often
presented to give the reader a choice of which form may be best suited for a
particular problem at hand. The equivalency of some of the LMIs in this
document may be straightforward to more experienced readers, but the authors
believe that some readers may benefit from the presentation of multiple
equivalent LMIs.
