  This paper focuses on investigating Stein's invariant shrinkage estimators
for large sample covariance matrices and precision matrices in high-dimensional
settings. We consider models that have nearly arbitrary population covariance
matrices, including those with potential spikes. By imposing mild technical
assumptions, we establish the asymptotic limits of the shrinkers for a wide
range of loss functions. A key contribution of this work, enabling the
derivation of the limits of the shrinkers, is a novel result concerning the
asymptotic distributions of the non-spiked eigenvectors of the sample
covariance matrices, which can be of independent interest.
