We perturb a real matrix $A$ of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our bounds, which extend existing lower-order expressions, demonstrate the potential increase in the smallest singular values, and represent a qualitative model for the increase in the small singular values after a matrix has been downcast to a lower arithmetic precision. Numerical experiments confirm the qualitative validity of this model and its ability to predict singular values changes in the presence of decreased arithmetic precision.

翻译：我们扰动一个满列秩的实矩阵 $A$，并基于范数绝对扰动推导了扰动矩阵最小奇异值的下界。这些下界扩展了现有的低阶表达式，揭示了最小奇异值可能增大的现象，并为矩阵降精度转换后小奇异值的增加提供了定性模型。数值实验验证了该模型的定性有效性及其在算术精度降低情境下预测奇异值变化的能力。