On the perturbation of an $L^2$-orthogonal projection

TL;DR

This paper develops new, sharper bounds for the perturbation of $L^2$-orthogonal projections onto subspaces, which are crucial in numerical analysis and various applied fields, supported by numerical examples.

Contribution

It introduces novel perturbation bounds for $L^2$-orthogonal projections that improve upon existing bounds in terms of sharpness and applicability.

Findings

New bounds involve upper and lower estimates.

Bounds are sharper than previous results.

Numerical examples validate the theoretical bounds.

Abstract

The $L^2$-orthogonal projection onto a subspace is an important mathematical tool, which has been widely applied in many fields such as linear least squares problems, eigenvalue problems, ill-posed problems, and randomized algorithms. In some numerical applications, the entries of a matrix will seldom be known exactly, so it is necessary to develop some bounds to characterize the effects of the uncertainties caused by matrix perturbation. In this paper, we establish new perturbation bounds for the $L^2$-orthogonal projection onto the column space of a matrix, which involve upper (lower) bounds and combined upper (lower) bounds. The new bounds contain some sharper counterparts of the existing ones. Numerical examples are also given to illustrate our theoretical results.