Mathematical Sciences: Stable Relations and Their Loci in Operator Variables

欧几里得空间中由多项式方程定义的曲线的抽象性质是很容易理解的。特别地，有限多项式集是一个稳定的交换未知数方程集，在以下意义上：任何近似解（多项式取小值的欧几里德空间中的点）将接近解集。当应用于算子值时，同一组多项式可能变得不稳定。事实上，即使变量中只允许有矩阵值，它们也经常变得不稳定；可能存在矩阵元组，其中多项式取很小的值，并且远离方程的矩阵值解集。这个项目的一个主要目标将是确定区分这种“幻影”近似解与真解的扰动的可计算不变量。在计算机模拟数学模型时，这些虚幻的近似解可能会产生误导。数值线性代数包不能用矩阵进行精确计算。例如，直到去年，还没有办法判断一个几乎满足所谓正态性方程的矩阵是否代表了实际正态矩阵的近似值。现在人们知道这是真的。由于线性代数的许多理论和应用在处理非正态矩阵时必须改变，这是一个重要的理论进步。它增加了我们对依赖计算机建模的信心。通过将稳定性结果添加到可以应用于线性代数的列表中，该项目应该增强这种信心。

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.