In this paper, we study the eigenvalues and eigenvectors of the spiked invariant multiplicative models when the randomness is from Haar matrices. We establish the limits of the outlier eigenvalues $\widehat{ \lambda}_i$ and the generalized components ($\langle \mathbf{ v}, \widehat{ \mathbf{ u}}_i \rangle$ for any deterministic vector $\mathbb{ v}$) of the outlier eigenvectors $\widehat{ \mathbf{ u}}_i$ with optimal convergence rates. Moreover, we prove that the non-outlier eigenvalues stick with those of the unspiked matrices and the non-outlier eigenvectors are delocalized. The results also hold near the so-called BBP transition and for degenerate spikes. On one hand, our results can be regarded as a refinement of the counterparts of [12] under additional regularity conditions. On the other hand, they can be viewed as an analog of [34] by replacing the random matrix with i.i.d. entries with Haar random matrix.
Categories: math.PR