Topics in matrix analysis
New estimates for Ritz vectors
Mathematics of Computation
SIAM Journal on Scientific Computing
New A Priori FEM Error Estimates for Eigenvalues
SIAM Journal on Numerical Analysis
Majorization for Changes in Angles Between Subspaces, Ritz Values, and Graph Laplacian Spectra
SIAM Journal on Matrix Analysis and Applications
Bounds on Changes in Ritz Values for a Perturbed Invariant Subspace of a Hermitian Matrix
SIAM Journal on Matrix Analysis and Applications
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The Rayleigh-Ritz (RR) method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator $A$. If the trial subspace is $A$-invariant, the Ritz values are exactly some of the eigenvalues of $A$. Given two subspaces $\mathcal{X}$ and $\mathcal{Y}$ of the same finite dimension, such that $\mathcal{X}$ is $A$-invariant, the absolute changes in the Ritz values of $A$ with respect to $\mathcal{X}$ compared to the Ritz values with respect to $\mathcal{Y}$ represent the RR absolute eigenvalue approximation error. Our first main result is a sharp majorization-type RR error bound in terms of the principal angles between $\mathcal{X}$ and $\mathcal{Y}$ for an arbitrary $A$-invariant $\mathcal{X}$, which was a conjecture in [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 548-559]. Second, we prove a novel type of RR error bound that deals with the products of the errors, rather than the sums. Third, we establish majorization bounds for the relative errors. We extend our bounds to the case $\dim\mathcal{X}\leq\dim\mathcal{Y}