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This paper presents an analytical performance investigation of an interference-limited double scattering multiple-input multiple-output (MIMO) channel employing optimum combining. Our main contribution is the derivation of the closed-form expressions for the cumulative distribution function (c.d.f.) and probability density function (p.d.f.) of the maximum eigenvalue of the resultant channel matrix after optimal combining when the transmit, receive and scattering correlation matrices are identities (hence, the channel is referred to as a Rayleigh-product channel). These results allow us to obtain the outage probability of the optimal combining system in a Rayleigh-product channel with co-channel interferences. Furthermore, we investigate, in depth, an important special case of double scattering channel, the keyhole channel, where based on various closed-form expressions for exact and asymptotic measures derived, we examine three key performance metrics, namely, the ergodic capacity, the outage performance and the symbol-error-rate (SER). The analytical results derived are validated by Monte-Carlo simulations.