  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.
