The notion of `stable rank' of a matrix is central to the analysis of randomized matrix algorithms, covariance estimation, deep neural networks, and recommender systems. We compare the properties of the stable rank and intrinsic dimension of real and complex matrices to those of the classical rank. Basic proofs and examples illustrate that the stable rank does not satisfy any of the fundamental rank properties, while the intrinsic dimension satisfies a few. In particular, the stable rank and intrinsic dimension of a submatrix can exceed those of the original matrix; adding a Hermitian positive semi-definite matrix can lower the intrinsic dimension of the sum; and multiplication by a nonsingular matrix can drastically change the stable rank and the intrinsic dimension. We generalize the concept of stable rank to the p-stable in any Schatten p-norm, thereby unifying the concepts of stable rank and intrinsic dimension: The stable rank is the 2-stable rank, while the intrinsic dimension is the 1-stable rank of a Hermitian positive semi-definite matrix. We derive sum and product inequalities for the pth root of the p-stable rank, and show that it is well-conditioned in the norm-wise absolute sense. The conditioning improves if the matrix and the perturbation are Hermitian positive semi-definite.

翻译：矩阵的“稳定秩”概念是分析随机化矩阵算法、协方差估计、深度神经网络以及推荐系统的核心。我们将实矩阵与复矩阵的稳定秩和内在维度的性质与经典秩的性质进行了比较。基本的证明与示例表明，稳定秩不满足任何基本的秩性质，而内在维度则满足其中少数几条。具体而言，子矩阵的稳定秩和内在维度可能超过原矩阵；添加一个Hermitian半正定矩阵可能降低和的内在维度；乘以一个非奇异矩阵可能显著改变稳定秩和内在维度。我们将稳定秩的概念推广至任意Schatten p-范数下的p-稳定秩，从而统一了稳定秩与内在维度的概念：稳定秩是2-稳定秩，而内在维度是Hermitian半正定矩阵的1-稳定秩。我们推导了p-稳定秩的p次方根的和与积不等式，并证明其在范数绝对意义下是良态的。若矩阵与扰动均为Hermitian半正定，则条件数会得到改善。