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 a 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：本文比较实矩阵与复矩阵的稳定秩及内在维度与经典秩的性质差异。通过基础证明与算例阐明：稳定秩不满足任何基本秩性质，而内在维度仅满足部分性质。具体而言，子矩阵的稳定秩与内在维度可能超过原矩阵；添加埃尔米特半正定矩阵可能降低和矩阵的内在维度；非奇异矩阵乘法可能显著改变稳定秩与内在维度。我们将稳定秩概念推广至沙滕p-范数下的p-稳定秩，从而统一稳定秩与内在维度的概念：稳定秩即2-稳定秩，而内在维度为埃尔米特半正定矩阵的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，证明其在范数绝对意义下具有良好条件性。当矩阵与扰动均为埃尔米特半正定时，条件性将得到进一步改善。