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Interference alignment in the K-user MIMO interference channel with constant channel coefficients is considered. A novel constructive method for finding the interference alignment solution is proposed for the case where the number of transmit antennas equals the number of receive antennas (NT = NR = N), the number of transmitter-receiver pairs equals K = N + 1, and all interference alignment multiplexing gains are one. The core of the method consists of solving an eigenvalue problem that incorporates the channel matrices of all interfering links. This procedure provides insight into the feasibility of signal vector spaces alignment schemes in finite dimensional MIMO interference channels.