The name is absent

described by (2.33) and (2.34). We now compose

T = Σ^-1∕²T*L*, and T~¹ = UZ^-^1/2,

and note that the transformed gramians

P = TPT* and Q = T~*QT~¹

are balanced and diagonal in the sense that

P= Q = Σ.

Moreover, they are the gramians of the transformed state, z = Tz, which itself is
governed by the transformed dynamical system

= Âz(t) + Bu(Z),    y (Z) = Cz(Z),

where A = TAT^-1, B = TB, C = CT^-1. Based on the decay of the singular values
in Σ, we can construct a reduced model by using only the к largest singular values.
This corresponds to approximating (2.35) with

ɪ = A₁₁ξ(Z) + Biu(Z), y(Z) = C₁ξ(Z),

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