# Spiked multiplicative random matrices and principal components

## TL;DR

This paper analyzes the spectral properties of spiked multiplicative random matrices with Haar randomness, providing precise limits for outlier eigenvalues and eigenvectors, and demonstrating eigenvalue sticking and eigenvector delocalization.

## Contribution

It offers new results on eigenvalue limits and eigenvector behavior in spiked Haar-invariant models, extending previous work and covering near the BBP transition and degenerate spikes.

## Key findings

- Outlier eigenvalues have established limits with optimal convergence.
- Eigenvectors associated with outliers are characterized with generalized components.
- Non-outlier eigenvalues stick to those of unspiked matrices, and eigenvectors are delocalized.

## Abstract

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.