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This paper studies a novel underlay MIMO cognitive radio (CR) system, where the instantaneous or statistical channel state information (CSI) of the interfering channels to the primary receivers (PRs) is completely unknown to the CR. For the single underlay receiver scenario, we assume a minimum information rate must be guaranteed on the CR main channel whose CSI is known at the CR transmitter. We first show that low-rank CR interference is preferable for improving the throughput of the PRs compared with spreading less power over more transmit dimensions. Based on this observation, we then propose a rank minimization CR transmission strategy assuming a minimum information rate must be guaranteed on the CR main channel. We propose a simple solution referred to as frugal waterfilling (FWF) that uses the least amount of power required to achieve the rate constraint with a minimum-rank transmit covariance matrix. We also present two heuristic approaches that have been used in prior work to transform rank minimization problems into convex optimization problems. The proposed schemes are then generalized to an underlay MIMO CR downlink network with multiple receivers. Finally, a theoretical analysis of the interference temperature and leakage rate outage probabilities at the PR is presented for Rayleigh fading channels.We demonstrate that the direct FWF solution leads to higher PR throughput even though it has higher interference "temperature (IT) compared with the heuristic methods and classic waterfilling, which calls into question the use of IT as a metric for CR interference., Comment: 11 pages