  This paper investigates the capability of a passive Reconfigurable
Intelligent Surface (RIS) to redistribute the singular values of point-to-point
Multiple-Input Multiple-Output (MIMO) channels for achieving power and rate
gains. We depart from the conventional Diagonal (D)-RIS with diagonal phase
shift matrix and adopt a Beyond Diagonal (BD) architecture that offers greater
wave manipulation flexibility through element-wise connections. Specifically,
we first provide shaping insights by characterizing the channel singular value
regions attainable by D-RIS and BD-RIS via a novel geodesic optimization.
Analytical singular value bounds are then derived to explore their shaping
limits in typical deployment scenarios. As a side product, we tackle
BD-RIS-aided MIMO rate maximization problem by a local-optimal Alternating
Optimization (AO) and a shaping-inspired low-complexity approach. Results show
that compared to D-RIS, BD-RIS significantly improves the dynamic range of all
channel singular values, the trade-off in manipulating them, and thus the
channel power and achievable rate. Those observations become more pronounced
when the number of RIS elements and MIMO dimensions increase. Of particular
interest, BD-RIS is shown to activate multi-stream transmission at lower
transmit power than D-RIS, hence achieving the asymptotic Degrees of Freedom
(DoF) at low Signal-to-Noise Ratio (SNR) thanks to its higher flexibility of
shaping the distribution of channel singular values.
