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We are interested in computing a simple eigenpair $(\lambda,{\bf x})$ of alarge non-Hermitian matrix $A$, by a general harmonic Rayleigh-Ritz projectionmethod. Given a search subspace $\mathcal{K}$ and a target point $\tau$, wefocus on the convergence of the harmonic Ritz vector $\widetilde{\bf x}$ andharmonic Ritz value $\widetilde{\lambda}$. In [{Z. Jia}, {\em The convergenceof harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritzvectors}, Math. Comput., 74 (2004), pp. 1441--1456.], Jia showed that for theconvergence of harmonic Ritz vector and harmonic Ritz value, it is essential toassume certain Rayleigh quotient matrix being {\it uniformly nonsingular} as$\angle({\bf x},\mathcal{K})\rightarrow 0$. However, this assumption can not beguaranteed theoretically for a general matrix $A$, and the Rayleigh quotientmatrix can be singular or near singular even if $\tau$ is not close to$\lambda$. In this paper, we abolish this constraint and derive new bounds forthe convergence of harmonic Rayleigh-Ritz projection methods. We show that asthe distance between ${\bf x}$ and $\mathcal{K}$ tends to zero and $\tau$ issatisfied with the so-called {\it uniform separation condition}, the harmonicRitz value converges, and the harmonic Ritz vector converges as$\frac{1}{\lambda-\tau}$ is well separated from other Ritz values of $(A-\tauI)^{-1}$ in the orthogonal complement of $(A-\tau I)\widetilde{\bf x}$ withrespect to $(A-\tau I)\mathcal{K}$.

On the convergence of harmonic Ritz vectors and harmonic Ritz values