$$
\begin{array}{l} \overleftarrow {\mathbf {x} _ {0}} = \mathcal {S} \mathbf {S} ^ {T} \overleftarrow {\mathbf {x} _ {1}} - 2 \zeta \mathbf {S} \mathbf {U} \boldsymbol {\Sigma} \mathbf {V} ^ {T} \mathbf {S} ^ {T} \overleftarrow {\mathbf {x} _ {1}} + 2 \zeta \mathbf {S} \mathbf {U} \boldsymbol {\Sigma} \mathbf {V} ^ {T} \overrightarrow {\mathbf {z} _ {0}} \\ = \mathcal {S} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x}} _ {1} - 2 \zeta \mathcal {S} \boldsymbol {U} \Sigma \boldsymbol {V} ^ {T} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x}} _ {1} + 2 \zeta \mathcal {S} \boldsymbol {U} \Sigma \boldsymbol {V} ^ {T} \overrightarrow {\boldsymbol {z}} _ {0} \\ = \mathcal {S} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x}} _ {1} - 2 (\mathcal {S} \boldsymbol {U}) \boldsymbol {D} (\boldsymbol {\zeta} _ {i}) (\mathcal {S} \boldsymbol {U}) ^ {T} \mathcal {S} \boldsymbol {U} \Sigma \boldsymbol {V} ^ {T} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x}} _ {1} + 2 (\mathcal {S} \boldsymbol {U}) \boldsymbol {D} (\boldsymbol {\zeta} _ {i}) (\mathcal {S} \boldsymbol {U}) ^ {T} \mathcal {S} \boldsymbol {U} \Sigma \boldsymbol {V} ^ {T} \overrightarrow {\boldsymbol {z}} _ {0} \\ \stackrel {(i)} {=} \mathcal {S} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x}} _ {1} - 2 (\mathcal {S} \boldsymbol {U}) \boldsymbol {D} (\zeta_ {i}) \boldsymbol {U} ^ {T} \boldsymbol {S} ^ {T} \mathcal {S} \boldsymbol {U} \Sigma \boldsymbol {V} ^ {T} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x}} _ {1} + 2 (\mathcal {S} \boldsymbol {U}) \boldsymbol {D} (\zeta_ {i}) \boldsymbol {U} ^ {T} \boldsymbol {S} ^ {T} \mathcal {S} \boldsymbol {U} \Sigma \boldsymbol {V} ^ {T} \overrightarrow {\boldsymbol {z}} _ {0} \\ \stackrel {(i i)} {=} \mathcal {S} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x}} _ {1} - 2 (\mathcal {S} \boldsymbol {U}) \boldsymbol {D} (\boldsymbol {\zeta} _ {i}) \boldsymbol {U} ^ {T} \boldsymbol {U} \Sigma \boldsymbol {U} ^ {T} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x}} _ {1} + 2 (\mathcal {S} \boldsymbol {U}) \boldsymbol {D} (\boldsymbol {\zeta} _ {i}) \boldsymbol {U} ^ {T} \boldsymbol {U} \Sigma \boldsymbol {U} ^ {T} \overrightarrow {\boldsymbol {z}} _ {0} \\ = \mathcal {S} S ^ {T} \overleftarrow {\boldsymbol {x}} _ {1} - 2 (\mathcal {S} U) \boldsymbol {D} (\boldsymbol {\zeta} _ {i}) \Sigma \boldsymbol {U} ^ {T} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x}} _ {1} + 2 (\mathcal {S} U) \boldsymbol {D} (\boldsymbol {\zeta} _ {i}) \Sigma \boldsymbol {U} ^ {T} \overrightarrow {\boldsymbol {z}} _ {0} \\ = \mathcal {S} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x}} _ {1} - 2 \mathcal {S} \mathcal {U} \mathcal {D} (\zeta_ {i}) \mathcal {D} (\sigma) \mathcal {U} ^ {T} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x}} _ {1} + 2 \mathcal {S} \mathcal {U} \mathcal {D} (\zeta_ {i}) \mathcal {D} (\sigma) \mathcal {U} ^ {T} \overrightarrow {\boldsymbol {z}} _ {0} \\ = \mathcal {S} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x} _ {1}} - 2 \mathcal {S} \boldsymbol {U} \boldsymbol {D} (\zeta_ {i} \odot \sigma) \boldsymbol {U} ^ {T} \mathcal {S} ^ {T} \overleftarrow {\boldsymbol {x} _ {1}} + 2 \mathcal {S} \boldsymbol {U} \boldsymbol {D} (\zeta_ {i} \odot \sigma) \boldsymbol {U} ^ {T} \overrightarrow {\boldsymbol {z} _ {0}}, \\ \end{array}
$$