This paper investigates the limits to which a passive Reconfigurable Intelligent Surface (RIS) can reshape a point-to-point Multiple-Input Multiple-Output (MIMO) in terms of singular values for improved wireless (e.g., rate and power) performance. We depart from the Diagonal (D) scattering matrix and adopt a Beyond Diagonal (BD) model that exploits element-wise connections for signal amplitude and phase manipulation. Specifically, analytical bounds are derived under popular RIS deployment scenarios to showcase the shaping potentials of BD-RIS on the channel Degrees of Freedom (DoF), singular value spread, power gain, and capacity. A numerical optimization method is then proposed for a broader range of shaping problems and invoked to characterize the achievable singular value region. As a side product, we tackle BD-RIS-aided MIMO rate maximization problem by a local-optimal Alternating Optimization (AO) and a low-complexity shaping-inspired approach. Results show that BD-RIS significantly improves the dynamic range of all channel singular values and the trade-off in manipulating them, resulting in enhanced power gain and achievable rate. Those benefits become more pronounced when the number of RIS elements or MIMO dimensions increase. Of particular interest, BD-RIS is shown to activate multi-stream transmission (hence achieving the asymptotic DoF) at much lower transmit power than D-RIS thanks to its singular value shaping proficiency.

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