Eigenvalue approximations of an Hermitian matrix $A$ can be found by the Rayleigh-Ritz method searching for eigenvalues of $X^H A X$ where the columns of the matrix $X$ form an orthogonal basis for a subspace $\mathcal{X}$, called the trial subspace. $X^H$ denotes the conjugate transpose of $X$. The eigenvalues of $X^H A X$ are called Ritz values. If the trial subspace is invariant with respect to $A$, the Ritz values are exactly some of the eigenvalues of $A$. The Ritz values do not depend on the particular choice of the basis. Thus, the authors consider the two subspaces $\mathcal{X}$ and $\mathcal{Y}$ of the same finite dimension, such that $\mathcal{X}$ is invariant with respect to $A$ and $\mathcal{Y}$ is a perturbation of $\mathcal{X}$. They state that the absolute changes in the Ritz values of $A$ with respect to $\mathcal{X}$ compared with the Ritz values with respect to $\mathcal{Y}$ give the absolute eigenvalue approximation error of the Rayleigh-Ritz method. A corresponding majorization error bound for this error was developed by {\it M. E. Argentati, A. V. Knyazev, C. C. Paige,} and {\it I. Panayotov} [ibid. 30, No.~2, 548‒559 (2008; Zbl 1171.15018)] in form of a conjecture. In the present paper the authors formulate their results for an Hermitian operator $A$ in a finite dimensional inner product space covering subjects of numerical linear algebra and approximation theory. The Rayleigh-Ritz method finds the stationary values. The authors prove sharp majorization Rayleigh-Ritz error bounds in terms of the principal angles between the subspaces $\mathcal{X}$ and $\mathcal{Y}$. In the above mentioned paper only some special cases of the conjecture have been proved for an Hermitian matrix $A$. Furthermore, a new multiplicative Rayleigh-Ritz error bound based on the product of errors is derived. Finally, the results are extended to the case $\dim\cal{X} \le \dim\cal{Y} < \infty$ in infinite dimensional Hilbert spaces and then applied to the finite element method (FEM), which is demonstrated for a membrane vibration problem, an eigenvalue problem in two dimensions.
Georg Hebermehl (Berlin)