Mathematical Sciences: Stable Relations and Their Loci in Operator Variables

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9531841 Loring The abstract properties of curves in Euclidean space defined by polynomial equations are well-understood. In particular, a finite set of polynomials is a stable set of equations in commuting unknowns in the following sense: any approximate solution (a point in Euclidean space where those polynomials take a small value) will be close to the solution set. The same set of polynomials can become unstable when applied to operator values. Indeed, they often become unstable even when only matrix values are allowed in the variables; there can be tuples of matrices upon which the polynomials take small values and which are far away from the matrix-valued solution set to the equations. A primary goal of this project will be to determine computable invariants for differentiating such "phantom" approximate solutions" from perturbations of true solutions. These phantom approximate solutions can be misleading in the context of computer simulations of mathematical models. A numerical linear algebra package cannot make exact calculations with matrices. Until last year, for example, there was no way to tell if a matrix which almost satisfied the so-called normality equation represented an approximation of an actual normal matrix. This is now known to be true. Since much of the theory, and applications, of linear algebra must be changed when working with non-normal matrices, this is an important theoretical advance. It adds to our confidence in our reliance on computer modeling. By adding to the list of stability results that can be applied to linear algebra, this project should enhance that confidence.