This paper addresses the problem of maximizing the capacity of a multiple-input multiple-output (MIMO) link assisted by a beyond-diagonal reconfigurable intelligent surface (BD-RIS). We maximize the capacity by alternately optimizing the transmit covariance matrix, and the BD-RIS scattering matrix, which, according to network theory, should be unitary and symmetric. These constraints make the optimization of BD-RIS more challenging than that of diagonal RIS. To find a stationary point of the capacity we maximize a sequence of quadratic problems in the manifold of unitary matrices. This leads to an efficient algorithm that always improves the capacity obtained by a diagonal RIS. Through simulation examples, we study the capacity improvement provided by a passive BD-RIS architecture over the conventional RIS model in which the phase shift matrix is diagonal.

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