An ill-conditioned overlap preserves exact Ritz values but amplifies coordinate errors

The vectors $q_{+}=(1,1)\mathrm{/}\sqrt{2}$ and $q_{-}=(1,-1)\mathrm{/}\sqrt{2}$ are orthonormal and satisfy $S_{\epsilon }{q}_{+}=(2-\epsilon )q_{+}$ and $S_{\epsilon }{q}_{-}=\epsilon q_{-}$. Both eigenvalues are positive, and their ratio is $(2-\epsilon )\mathrm{/}\epsilon$.

Let $S_{\epsilon }=L{L}^{\ast }$ be its Cholesky factorization. By [F1], $y=L^{\ast }c$ preserves the exact generalized eigenvalues and Rayleigh quotient. For any matrix $A$, $\mathrm{\parallel }A{\mathrm{\parallel }}_2^2={\max }_{\mathrm{\parallel }x{\mathrm{\parallel }}_2=1}{x}^{\ast }{A}^{\ast }Ax$, which equals the largest eigenvalue of the positive Hermitian matrix $A^{\ast }A$ by [F2]. Since $(L^{-1}{)}^{\ast }{L}^{-1}=L^{-\ast }{L}^{-1}=S_{\epsilon }^{-1}$, its eigenvalues are $1\mathrm{/}\epsilon$ and $1\mathrm{/}(2-\epsilon )$, so $\mathrm{\parallel }{L}^{-1}{\mathrm{\parallel }}_2={\epsilon }^{-1\mathrm{/}2}$. For $A=L^{-\ast }$, $A^{\ast }A=L^{-1}{L}^{-\ast }=L^{-1}{S}_{\epsilon }^{-1}L$, which is similar to $S_{\epsilon }^{-1}$; hence $\mathrm{\parallel }{L}^{-\ast }{\mathrm{\parallel }}_2={\epsilon }^{-1\mathrm{/}2}$ as well. The operator norm is attained on a unit vector in this finite-dimensional space, so a unit perturbation in a maximizing direction is amplified by this factor when converted back to coefficient coordinates.

As $\epsilon \downarrow 0$, the condition number and coordinate amplification diverge, although the exact whitening identities in [F1] continue to hold for each $\epsilon >0$. This is a conditioning observation; it gives no finite-precision energy-error estimate without a specified perturbation model.