Spiked multiplicative random matrices and principal components. (English) Zbl 1525.15027
Authors’ 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 \(\mathbf{v})\) 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 S. T. Belinschi et al. [Ann. Probab. 45, No. 6A, 3571–3625 (2017; Zbl 1412.60014)] under additional regularity conditions. On the other hand, they can be viewed as an analog of X. Ding and F. Yang [Ann. Stat. 49, No. 2, 1113–1138 (2021; Zbl 1469.15035)] by replacing the random matrix with i.i.d. entries with Haar random matrix.
Reviewer: Sho Matsumoto (Kagoshima)
MSC:
| 15B52 | Random matrices (algebraic aspects) |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 46L54 | Free probability and free operator algebras |
| 60B20 | Random matrices (probabilistic aspects) |
| 62H12 | Estimation in multivariate analysis |
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