  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.
