Manipulating Matrix Inequalities Automatically

Matrix inequalities have come to be extremely important in systems engineering in the past decade. This is because many systems problems convert directly into matrix inequalities. Matrix inequalities take the form of a list of requirements that polynomials or rational functions of matrices be positive semide nite. Of course while some engineering problems present rational functions which are well behaved, many other problems present rational functions which are badly behaved. Thus taking the list of functions which a design problem presents and converting these to a nice form, or at least checking if they already have or do not have a nice form is a major enterprise. Since matrix multiplication is not commutative, one sees much e ort going into calculations (by hand) on noncommutative rational functions. A major goal in systems engineering is to convert, if possible,\noncommutative inequalities" to equivalent Linear Noncommutative Inequalities (e ectively to Linear Matrix Inequalities, to LMI's). This survey concerns e orts to process \noncommutative inequalities" using computer algebra. The most basic e orts, such as determining when noncommutative polynomials are positive, convex, convertible to noncommutative LMI's, transformable to convex inequalities, etc., force one to a rich area of undeveloped mathematics. OUTLINE To Commute or Not Commute: An Homage to Formulas which Scale Elegantly Noncommutative Inequalities Behave Better than Commutative Ones Which Sets have LMI Representations? LMI Representations for Sets which Automatically Scale Noncommutative Convexity