Eigenvector overlaps of random covariance matrices and their submatrices

Abstract We consider the singular vectors of any m × n submatrix of a rectangular M × N Gaussian matrix and study their asymptotic overlaps with those of the full matrix, in the macroscopic regime where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mstyle scriptlevel="0"/> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> as well as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> converge to fixed ratios. Our method makes use of the dynamics of the singular vectors and of specific resolvents when the matrix coefficients follow Brownian trajectories. We obtain explicit forms for the limiting rescaled mean squared overlaps for right and left singular vectors in the bulk of both spectra, for any initial matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mstyle scriptlevel="0"/> </mml:mrow> </mml:math> . When it is null, this corresponds to the Marchenko–Pastur setup for covariance matrices, and our formulas simplify into Cauchy-like functions.