THE ORDER IS NOW THE FOLLOWING: B_{1} < B_{2} < B_{1}^{T} < B_{2}^{T} < C_{1} < C_{2} < C_{1}^{T} < C_{2}^{T} < D_{11} < D_{12} < D_{21} < D_{22} < D_{11}^{T} < D_{12}^{T} < D_{21}^{T} < D_{22}^{T} < A_{11} < A_{12} < A_{21} < A_{22} < A_{11}^{T} < A_{12}^{T} < A_{21}^{T} < A_{22}^{T} ≪ Σ < Σ^{−1} < Σ^{T} < Σ^{−1T} ≪ P_{11} < P_{11}^{−1} < P_{11}^{T} < P_{11}^{−1T} < P_{22} < P_{22}^{−1} < P_{22}^{T} < P_{22}^{−1T} < P_{33} < P_{33}^{−1} < P_{33}^{T} < P_{33}^{−1T} ≪ P_{12} < P_{12}^{T} < P_{13} < P_{13}^{T} < P_{21} < P_{21}^{T} < P_{23} < P_{23}^{T} < P_{31} < P_{31}^{T} < P_{32} < P_{32}^{T}

YOUR SESSION HAS DIGESTED THE FOLLOWING RELATIONS

THE FOLLOWING VARIABLES HAVE BEEN SOLVED FOR: {D_{11}, D_{12}, D_{21}, D_{22}, P_{13}, P_{21}, P_{23}, P_{31}, P_{32}, P_{33}, P_{33}^{−1}, P_{33}^{−1T} , D_{11}^{T}, D_{12}^{T}, D_{21}^{T}, D_{22}^{T}, P_{11}^{T}, P_{13}^{T}, P_{21}^{T}, P_{22}^{T}, P_{23}^{T}, P_{31}^{T}, P_{32}^{T}, P_{33}^{T}, Σ^{T} , Σ^{−1T} } The corresponding rules are the following: D_{11} → 0 D_{12} → 2 D_{21} → 2 D_{22} → 0 P_{13} → 0 P_{21} → P_{12}^{T} P_{23} → 0 P_{31} → 0 P_{32} → 0 P_{33} → 2 P_{33}^{−1} → \frac{1}{2} P_{33}^{−1T} → \frac{1}{2} D_{11}^{T} → 0 D_{12}^{T} → 2 D_{21}^{T} → 2 D_{22}^{T} → 0 P_{11}^{T} → P_{11} P_{13}^{T} → 0 P_{21}^{T} → P_{12} P_{22}^{T} → P_{22} P_{23}^{T} → 0 P_{31}^{T} → 0 P_{32}^{T} → 0 P_{33}^{T} → 2 Σ^{T} → Σ Σ^{−1T} → Σ^{−1}

The expressions with unknown variables {} and knowns {A_{11}, A_{12}, A_{21}, A_{22}, B_{1}, B_{2}, C_{1}, C_{2}, A_{11}^{T}, A_{12}^{T}, A_{21}^{T}, A_{22}^{T}, B_{1}^{T} , B_{2}^{T} , C_{1}^{T} , C_{2}^{T} } B_{2} B_{2}^{T} → −1 A_{22} − A_{22}^{T}

C_{2}^{T} C_{2} → −1 A_{22} − A_{22}^{T}

A_{21} B_{1} B_{2}^{T} → −1 A_{21} A_{12} + A_{12}^{T} A_{21}^{T} + B_{2} B_{1}^{T} A_{21}^{T}

A_{21} C_{1}^{T} C_{2} → −1 A_{21} A_{21}^{T} + A_{12}^{T} A_{12} + B_{2} B_{1}^{T} A_{12}

A_{12}^{T} B_{1} B_{2}^{T} → A_{21} A_{21}^{T} − A_{12}^{T} A_{12} + C_{2}^{T} C_{1} A_{21}^{T}

A_{12}^{T} C_{1}^{T} C_{2} → A_{21} A_{12} − A_{12}^{T} A_{21}^{T} + C_{2}^{T} C_{1} A_{12}

A_{21} A_{11} B_{1} B_{2}^{T} → −1 A_{21} A_{11} A_{12} − A_{12}^{T} A_{11}^{T} A_{21}^{T} + A_{21} B_{1} B_{1}^{T} A_{21}^{T} − B_{2} B_{1}^{T} A_{11}^{T} A_{21}^{T}

A_{21} A_{11} C_{1}^{T} C_{2} → −1 A_{21} A_{11} A_{21}^{T} − A_{12}^{T} A_{11}^{T} A_{12} + A_{21} B_{1} B_{1}^{T} A_{12} − B_{2} B_{1}^{T} A_{11}^{T} A_{12}

A_{21} A_{11}^{T} B_{1} B_{2}^{T} → −1 A_{21} A_{11}^{T} A_{12} − A_{12}^{T} A_{11} A_{21}^{T} + A_{21} C_{1}^{T} C_{1} A_{21}^{T} − B_{2} B_{1}^{T} A_{11} A_{21}^{T}

A_{21} A_{11}^{T} C_{1}^{T} C_{2} → −1 A_{21} A_{11}^{T} A_{21}^{T} − A_{12}^{T} A_{11} A_{12} + A_{21} C_{1}^{T} C_{1} A_{12} − B_{2} B_{1}^{T} A_{11} A_{12}

A_{12}^{T} A_{11} B_{1} B_{2}^{T} → −1 A_{21} A_{11}^{T} A_{21}^{T} − A_{12}^{T} A_{11} A_{12} + A_{12}^{T} B_{1} B_{1}^{T} A_{21}^{T} − C_{2}^{T} C_{1} A_{11}^{T} A_{21}^{T}

A_{12}^{T} A_{11} C_{1}^{T} C_{2} → −1 A_{21} A_{11}^{T} A_{12} − A_{12}^{T} A_{11} A_{21}^{T} + A_{12}^{T} B_{1} B_{1}^{T} A_{12} − C_{2}^{T} C_{1} A_{11}^{T} A_{12}

A_{12}^{T} A_{11}^{T} B_{1} B_{2}^{T} → −1 A_{21} A_{11} A_{21}^{T} − A_{12}^{T} A_{11}^{T} A_{12} + A_{12}^{T} C_{1}^{T} C_{1} A_{21}^{T} − C_{2}^{T} C_{1} A_{11} A_{21}^{T}

A_{12}^{T} A_{11}^{T} C_{1}^{T} C_{2} → −1 A_{21} A_{11} A_{12} − A_{12}^{T} A_{11}^{T} A_{21}^{T} + A_{12}^{T} C_{1}^{T} C_{1} A_{12} − C_{2}^{T} C_{1} A_{11} A_{12}

{A_{21} A_{11} A_{11} B_{1} B_{2}^{T} → \atop −1 A_{21} A_{11} A_{11} A_{12} +A_{12}^{T} A_{11}^{T} A_{11}^{T} A_{21}^{T} +A_{21} A_{11} B_{1} B_{1}^{T} A_{21}^{T} −A_{21} B_{1} B_{1}^{T} A_{11}^{T} A_{21}^{T} +B_{2} B_{1}^{T} A_{11}^{T} A_{11}^{T} A_{21}^{T}}

{A_{21} A_{11} A_{11} C_{1}^{T} C_{2} → \atop −1 A_{21} A_{11} A_{11} A_{21}^{T} +A_{12}^{T} A_{11}^{T} A_{11}^{T} A_{12} +A_{21} A_{11} B_{1} B_{1}^{T} A_{12} −A_{21} B_{1} B_{1}^{T} A_{11}^{T} A_{12} +B_{2} B_{1}^{T} A_{11}^{T} A_{11}^{T} A_{12}}

{A_{21} A_{11} A_{11}^{T} B_{1} B_{2}^{T} → \atop −1 A_{21} A_{11} A_{11}^{T} A_{12} +A_{12}^{T} A_{11}^{T} A_{11} A_{21}^{T} +A_{21} A_{11} C_{1}^{T} C_{1} A_{21}^{T} −A_{21} B_{1} B_{1}^{T} A_{11} A_{21}^{T} +B_{2} B_{1}^{T} A_{11}^{T} A_{11} A_{21}^{T}}

{A_{21} A_{11} A_{11}^{T} C_{1}^{T} C_{2} → \atop −1 A_{21} A_{11} A_{11}^{T} A_{21}^{T} +A_{12}^{T} A_{11}^{T} A_{11} A_{12} +A_{21} A_{11} C_{1}^{T} C_{1} A_{12} −A_{21} B_{1} B_{1}^{T} A_{11} A_{12} +B_{2} B_{1}^{T} A_{11}^{T} A_{11} A_{12}}

{A_{21} A_{11}^{T} A_{11} B_{1} B_{2}^{T} → \atop −1 A_{21} A_{11}^{T} A_{11} A_{12} +A_{12}^{T} A_{11} A_{11}^{T} A_{21}^{T} +A_{21} A_{11}^{T} B_{1} B_{1}^{T} A_{21}^{T} −A_{21} C_{1}^{T} C_{1} A_{11}^{T} A_{21}^{T} +B_{2} B_{1}^{T} A_{11} A_{11}^{T} A_{21}^{T}}

{A_{21} A_{11}^{T} A_{11} C_{1}^{T} C_{2} → \atop −1 A_{21} A_{11}^{T} A_{11} A_{21}^{T} +A_{12}^{T} A_{11} A_{11}^{T} A_{12} +A_{21} A_{11}^{T} B_{1} B_{1}^{T} A_{12} −A_{21} C_{1}^{T} C_{1} A_{11}^{T} A_{12} +B_{2} B_{1}^{T} A_{11} A_{11}^{T} A_{12}}

{A_{21} A_{11}^{T} A_{11}^{T} B_{1} B_{2}^{T} → \atop −1 A_{21} A_{11}^{T} A_{11}^{T} A_{12} +A_{12}^{T} A_{11} A_{11} A_{21}^{T} +A_{21} A_{11}^{T} C_{1}^{T} C_{1} A_{21}^{T} −A_{21} C_{1}^{T} C_{1} A_{11} A_{21}^{T} +B_{2} B_{1}^{T} A_{11} A_{11} A_{21}^{T}}

{A_{21} A_{11}^{T} A_{11}^{T} C_{1}^{T} C_{2} → \atop −1 A_{21} A_{11}^{T} A_{11}^{T} A_{21}^{T} +A_{12}^{T} A_{11} A_{11} A_{12} +A_{21} A_{11}^{T} C_{1}^{T} C_{1} A_{12} −A_{21} C_{1}^{T} C_{1} A_{11} A_{12} +B_{2} B_{1}^{T} A_{11} A_{11} A_{12}}

{A_{12}^{T} A_{11} A_{11} B_{1} B_{2}^{T} → A_{21} A_{11}^{T} A_{11}^{T} A_{21}^{T} −A_{12}^{T} A_{11} A_{11} A_{12} +A_{12}^{T} A_{11} B_{1} B_{1}^{T} A_{21}^{T} −A_{12}^{T} B_{1} B_{1}^{T} A_{11}^{T} A_{21}^{T} + \atop C_{2}^{T} C_{1} A_{11}^{T} A_{11}^{T} A_{21}^{T}}

{A_{12}^{T} A_{11} A_{11} C_{1}^{T} C_{2} → A_{21} A_{11}^{T} A_{11}^{T} A_{12} −A_{12}^{T} A_{11} A_{11} A_{21}^{T} +A_{12}^{T} A_{11} B_{1} B_{1}^{T} A_{12} −A_{12}^{T} B_{1} B_{1}^{T} A_{11}^{T} A_{12} + \atop C_{2}^{T} C_{1} A_{11}^{T} A_{11}^{T} A_{12}}

{A_{12}^{T} A_{11} A_{11}^{T} B_{1} B_{2}^{T} → A_{21} A_{11}^{T} A_{11} A_{21}^{T} −A_{12}^{T} A_{11} A_{11}^{T} A_{12} +A_{12}^{T} A_{11} C_{1}^{T} C_{1} A_{21}^{T} −A_{12}^{T} B_{1} B_{1}^{T} A_{11} A_{21}^{T} + \atop C_{2}^{T} C_{1} A_{11}^{T} A_{11} A_{21}^{T}}

{A_{12}^{T} A_{11} A_{11}^{T} C_{1}^{T} C_{2} → A_{21} A_{11}^{T} A_{11} A_{12} −A_{12}^{T} A_{11} A_{11}^{T} A_{21}^{T} +A_{12}^{T} A_{11} C_{1}^{T} C_{1} A_{12} −A_{12}^{T} B_{1} B_{1}^{T} A_{11} A_{12} + \atop C_{2}^{T} C_{1} A_{11}^{T} A_{11} A_{12}}

{A_{12}^{T} A_{11}^{T} A_{11} B_{1} B_{2}^{T} → A_{21} A_{11} A_{11}^{T} A_{21}^{T} −A_{12}^{T} A_{11}^{T} A_{11} A_{12} +A_{12}^{T} A_{11}^{T} B_{1} B_{1}^{T} A_{21}^{T} −A_{12}^{T} C_{1}^{T} C_{1} A_{11}^{T} A_{21}^{T} + \atop C_{2}^{T} C_{1} A_{11} A_{11}^{T} A_{21}^{T}}

{A_{12}^{T} A_{11}^{T} A_{11} C_{1}^{T} C_{2} → A_{21} A_{11} A_{11}^{T} A_{12} −A_{12}^{T} A_{11}^{T} A_{11} A_{21}^{T} +A_{12}^{T} A_{11}^{T} B_{1} B_{1}^{T} A_{12} −A_{12}^{T} C_{1}^{T} C_{1} A_{11}^{T} A_{12} + \atop C_{2}^{T} C_{1} A_{11} A_{11}^{T} A_{12}}

{A_{12}^{T} A_{11}^{T} A_{11}^{T} B_{1} B_{2}^{T} → A_{21} A_{11} A_{11} A_{21}^{T} −A_{12}^{T} A_{11}^{T} A_{11}^{T} A_{12} +A_{12}^{T} A_{11}^{T} C_{1}^{T} C_{1} A_{21}^{T} −A_{12}^{T} C_{1}^{T} C_{1} A_{11} A_{21}^{T} + \atop C_{2}^{T} C_{1} A_{11} A_{11} A_{21}^{T}}

{A_{12}^{T} A_{11}^{T} A_{11}^{T} C_{1}^{T} C_{2} → A_{21} A_{11} A_{11} A_{12} −A_{12}^{T} A_{11}^{T} A_{11}^{T} A_{21}^{T} +A_{12}^{T} A_{11}^{T} C_{1}^{T} C_{1} A_{12} −A_{12}^{T} C_{1}^{T} C_{1} A_{11} A_{12} + \atop C_{2}^{T} C_{1} A_{11} A_{11} A_{12}}

USER CREATIONS APPEAR BELOW

SOME RELATIONS WHICH APPEAR BELOW MAY BE UNDIGESTED

THE FOLLOWING VARIABLES HAVE NOT BEEN SOLVED FOR: {A_{11}, A_{12}, A_{21}, A_{22}, B_{1}, B_{2}, C_{1}, C_{2}, P_{11}, P_{12}, P_{22}, Σ, P_{11}^{−1}, P_{22}^{−1}, Σ^{−1}, P_{11}^{−1T} , P_{22}^{−1T} , A_{11}^{T}, A_{12}^{T}, A_{21}^{T}, A_{22}^{T}, B_{1}^{T} , B_{2}^{T} , C_{1}^{T} , C_{2}^{T} , P_{12}^{T}}

The expressions with unknown variables {P_{11}} and knowns {A_{11}, B_{1}, A_{11}^{T}, B_{1}^{T} } P_{11} A_{11}^{T} → −1 A_{11} P_{11} − B_{1} B_{1}^{T}

The expressions with unknown variables {P_{12}} and knowns {A_{11}, A_{11}^{T}, C_{1}^{T} } P_{12} A_{11}^{T} → −1 A_{11} P_{12} P_{12} C_{1}^{T} → 0

The expressions with unknown variables {Σ} and knowns {A_{11}, A_{12}, A_{21}, B_{1}, B_{2}, C_{1}, C_{2}, A_{11}^{T}, A_{12}^{T}, A_{21}^{T}, B_{1}^{T} , B_{2}^{T} , C_{1}^{T} , C_{2}^{T} } A_{21} Σ → −1 B_{2} B_{1}^{T} − A_{12}^{T} Σ A_{11} → −1 A_{11}^{T} Σ − C_{1}^{T} C_{1} Σ A_{12} → −1 C_{1}^{T} C_{2} − A_{21}^{T} Σ A_{11}^{T} → −1 A_{11} Σ − B_{1} B_{1}^{T} Σ A_{21}^{T} → −1 A_{12} − B_{1} B_{2}^{T} A_{12}^{T} Σ → −1 A_{21} − C_{2}^{T} C_{1} A_{21} A_{11} Σ → A_{12}^{T} A_{11}^{T} − A_{21} B_{1} B_{1}^{T} + B_{2} B_{1}^{T} A_{11}^{T} A_{21} A_{11}^{T} Σ → A_{12}^{T} A_{11} − A_{21} C_{1}^{T} C_{1} + B_{2} B_{1}^{T} A_{11} A_{12}^{T} A_{11} Σ → A_{21} A_{11}^{T} − A_{12}^{T} B_{1} B_{1}^{T} + C_{2}^{T} C_{1} A_{11}^{T} A_{12}^{T} A_{11}^{T} Σ → A_{21} A_{11} − A_{12}^{T} C_{1}^{T} C_{1} + C_{2}^{T} C_{1} A_{11} A_{21} A_{11} A_{11} Σ → −1 A_{12}^{T} A_{11}^{T} A_{11}^{T} − A_{21} A_{11} B_{1} B_{1}^{T} + A_{21} B_{1} B_{1}^{T} A_{11}^{T} − B_{2} B_{1}^{T} A_{11}^{T} A_{11}^{T} A_{21} A_{11} A_{11}^{T} Σ → −1 A_{12}^{T} A_{11}^{T} A_{11} − A_{21} A_{11} C_{1}^{T} C_{1} + A_{21} B_{1} B_{1}^{T} A_{11} − B_{2} B_{1}^{T} A_{11}^{T} A_{11} A_{21} A_{11}^{T} A_{11} Σ → −1 A_{12}^{T} A_{11} A_{11}^{T} − A_{21} A_{11}^{T} B_{1} B_{1}^{T} + A_{21} C_{1}^{T} C_{1} A_{11}^{T} − B_{2} B_{1}^{T} A_{11} A_{11}^{T} A_{21} A_{11}^{T} A_{11}^{T} Σ → −1 A_{12}^{T} A_{11} A_{11} − A_{21} A_{11}^{T} C_{1}^{T} C_{1} + A_{21} C_{1}^{T} C_{1} A_{11} − B_{2} B_{1}^{T} A_{11} A_{11} A_{12}^{T} A_{11} A_{11} Σ → −1 A_{21} A_{11}^{T} A_{11}^{T} − A_{12}^{T} A_{11} B_{1} B_{1}^{T} + A_{12}^{T} B_{1} B_{1}^{T} A_{11}^{T} − C_{2}^{T} C_{1} A_{11}^{T} A_{11}^{T} A_{12}^{T} A_{11} A_{11}^{T} Σ → −1 A_{21} A_{11}^{T} A_{11} − A_{12}^{T} A_{11} C_{1}^{T} C_{1} + A_{12}^{T} B_{1} B_{1}^{T} A_{11} − C_{2}^{T} C_{1} A_{11}^{T} A_{11} A_{12}^{T} A_{11}^{T} A_{11} Σ → −1 A_{21} A_{11} A_{11}^{T} − A_{12}^{T} A_{11}^{T} B_{1} B_{1}^{T} + A_{12}^{T} C_{1}^{T} C_{1} A_{11}^{T} − C_{2}^{T} C_{1} A_{11} A_{11}^{T} A_{12}^{T} A_{11}^{T} A_{11}^{T} Σ → −1 A_{21} A_{11} A_{11} − A_{12}^{T} A_{11}^{T} C_{1}^{T} C_{1} + A_{12}^{T} C_{1}^{T} C_{1} A_{11} − C_{2}^{T} C_{1} A_{11} A_{11}

{A_{21} A_{11} A_{11} A_{11} Σ → \atop A_{12}^{T} A_{11}^{T} A_{11}^{T} A_{11}^{T} −A_{21} A_{11} A_{11} B_{1} B_{1}^{T} +A_{21} A_{11} B_{1} B_{1}^{T} A_{11}^{T} −A_{21} B_{1} B_{1}^{T} A_{11}^{T} A_{11}^{T} +B_{2} B_{1}^{T} A_{11}^{T} A_{11}^{T} A_{11}^{T}}

{A_{21} A_{11} A_{11} A_{11}^{T} Σ → \atop A_{12}^{T} A_{11}^{T} A_{11}^{T} A_{11} −A_{21} A_{11} A_{11} C_{1}^{T} C_{1} +A_{21} A_{11} B_{1} B_{1}^{T} A_{11} −A_{21} B_{1} B_{1}^{T} A_{11}^{T} A_{11} +B_{2} B_{1}^{T} A_{11}^{T} A_{11}^{T} A_{11}}

{A_{21} A_{11} A_{11}^{T} A_{11} Σ → \atop A_{12}^{T} A_{11}^{T} A_{11} A_{11}^{T} −A_{21} A_{11} A_{11}^{T} B_{1} B_{1}^{T} +A_{21} A_{11} C_{1}^{T} C_{1} A_{11}^{T} −A_{21} B_{1} B_{1}^{T} A_{11} A_{11}^{T} +B_{2} B_{1}^{T} A_{11}^{T} A_{11} A_{11}^{T}}

{A_{21} A_{11} A_{11}^{T} A_{11}^{T} Σ → \atop A_{12}^{T} A_{11}^{T} A_{11} A_{11} − A_{21} A_{11} A_{11}^{T} C_{1}^{T} C_{1} + A_{21} A_{11} C_{1}^{T} C_{1} A_{11} − A_{21} B_{1} B_{1}^{T} A_{11} A_{11} + B_{2} B_{1}^{T} A_{11}^{T} A_{11} A_{11}}

{A_{21} A_{11}^{T} A_{11} A_{11} Σ → \atop A_{12}^{T} A_{11} A_{11}^{T} A_{11}^{T} −A_{21} A_{11}^{T} A_{11} B_{1} B_{1}^{T} +A_{21} A_{11}^{T} B_{1} B_{1}^{T} A_{11}^{T} −A_{21} C_{1}^{T} C_{1} A_{11}^{T} A_{11}^{T} +B_{2} B_{1}^{T} A_{11} A_{11}^{T} A_{11}^{T}}

{A_{21} A_{11}^{T} A_{11} A_{11}^{T} Σ → \atop A_{12}^{T} A_{11} A_{11}^{T} A_{11} − A_{21} A_{11}^{T} A_{11} C_{1}^{T} C_{1} + A_{21} A_{11}^{T} B_{1} B_{1}^{T} A_{11} − A_{21} C_{1}^{T} C_{1} A_{11}^{T} A_{11} + B_{2} B_{1}^{T} A_{11} A_{11}^{T} A_{11}}

{A_{21} A_{11}^{T} A_{11}^{T} A_{11} Σ → \atop A_{12}^{T} A_{11} A_{11} A_{11}^{T} − A_{21} A_{11}^{T} A_{11}^{T} B_{1} B_{1}^{T} + A_{21} A_{11}^{T} C_{1}^{T} C_{1} A_{11}^{T} − A_{21} C_{1}^{T} C_{1} A_{11} A_{11}^{T} + B_{2} B_{1}^{T} A_{11} A_{11} A_{11}^{T}}

{A_{21} A_{11}^{T} A_{11}^{T} A_{11}^{T} Σ → \atop A_{12}^{T} A_{11} A_{11} A_{11} − A_{21} A_{11}^{T} A_{11}^{T} C_{1}^{T} C_{1} + A_{21} A_{11}^{T} C_{1}^{T} C_{1} A_{11} − A_{21} C_{1}^{T} C_{1} A_{11} A_{11} + B_{2} B_{1}^{T} A_{11} A_{11} A_{11}}

{A_{12}^{T} A_{11} A_{11} A_{11} Σ → \atop A_{21} A_{11}^{T} A_{11}^{T} A_{11}^{T} −A_{12}^{T} A_{11} A_{11} B_{1} B_{1}^{T} +A_{12}^{T} A_{11} B_{1} B_{1}^{T} A_{11}^{T} −A_{12}^{T} B_{1} B_{1}^{T} A_{11}^{T} A_{11}^{T} +C_{2}^{T} C_{1} A_{11}^{T} A_{11}^{T} A_{11}^{T}}

{A_{12}^{T} A_{11} A_{11} A_{11}^{T} Σ → \atop A_{21} A_{11}^{T} A_{11}^{T} A_{11} − A_{12}^{T} A_{11} A_{11} C_{1}^{T} C_{1} + A_{12}^{T} A_{11} B_{1} B_{1}^{T} A_{11} − A_{12}^{T} B_{1} B_{1}^{T} A_{11}^{T} A_{11} + C_{2}^{T} C_{1} A_{11}^{T} A_{11}^{T} A_{11}}

{A_{12}^{T} A_{11} A_{11}^{T} A_{11} Σ → \atop A_{21} A_{11}^{T} A_{11} A_{11}^{T} − A_{12}^{T} A_{11} A_{11}^{T} B_{1} B_{1}^{T} + A_{12}^{T} A_{11} C_{1}^{T} C_{1} A_{11}^{T} − A_{12}^{T} B_{1} B_{1}^{T} A_{11} A_{11}^{T} + C_{2}^{T} C_{1} A_{11}^{T} A_{11} A_{11}^{T}}

{A_{12}^{T} A_{11} A_{11}^{T} A_{11}^{T} Σ → \atop A_{21} A_{11}^{T} A_{11} A_{11} − A_{12}^{T} A_{11} A_{11}^{T} C_{1}^{T} C_{1} + A_{12}^{T} A_{11} C_{1}^{T} C_{1} A_{11} − A_{12}^{T} B_{1} B_{1}^{T} A_{11} A_{11} + C_{2}^{T} C_{1} A_{11}^{T} A_{11} A_{11}}

{A_{12}^{T} A_{11}^{T} A_{11} A_{11} Σ → \atop A_{21} A_{11} A_{11}^{T} A_{11}^{T} − A_{12}^{T} A_{11}^{T} A_{11} B_{1} B_{1}^{T} + A_{12}^{T} A_{11}^{T} B_{1} B_{1}^{T} A_{11}^{T} − A_{12}^{T} C_{1}^{T} C_{1} A_{11}^{T} A_{11}^{T} + C_{2}^{T} C_{1} A_{11} A_{11}^{T} A_{11}^{T}}

{A_{12}^{T} A_{11}^{T} A_{11} A_{11}^{T} Σ → \atop A_{21} A_{11} A_{11}^{T} A_{11} − A_{12}^{T} A_{11}^{T} A_{11} C_{1}^{T} C_{1} + A_{12}^{T} A_{11}^{T} B_{1} B_{1}^{T} A_{11} − A_{12}^{T} C_{1}^{T} C_{1} A_{11}^{T} A_{11} + C_{2}^{T} C_{1} A_{11} A_{11}^{T} A_{11}}

{A_{12}^{T} A_{11}^{T} A_{11}^{T} A_{11} Σ → \atop A_{21} A_{11} A_{11} A_{11}^{T} − A_{12}^{T} A_{11}^{T} A_{11}^{T} B_{1} B_{1}^{T} + A_{12}^{T} A_{11}^{T} C_{1}^{T} C_{1} A_{11}^{T} − A_{12}^{T} C_{1}^{T} C_{1} A_{11} A_{11}^{T} + C_{2}^{T} C_{1} A_{11} A_{11} A_{11}^{T}}

{A_{12}^{T} A_{11}^{T} A_{11}^{T} A_{11}^{T} Σ → \atop A_{21} A_{11} A_{11} A_{11} − A_{12}^{T} A_{11}^{T} A_{11}^{T} C_{1}^{T} C_{1} + A_{12}^{T} A_{11}^{T} C_{1}^{T} C_{1} A_{11} − A_{12}^{T} C_{1}^{T} C_{1} A_{11} A_{11} + C_{2}^{T} C_{1} A_{11} A_{11} A_{11}}

The expressions with unknown variables {P_{22}^{−1}} and knowns {A_{11}, C_{1}, A_{11}^{T}, C_{1}^{T} } P_{22}^{−1} A_{11} → −1 A_{11}^{T} P_{22}^{−1} − C_{1}^{T} C_{1} The expressions with unknown variables {Σ^{−1}} and knowns {A_{11}, A_{12}, A_{21}, B_{1}, B_{2}, C_{1}, C_{2}, A_{11}^{T}, A_{12}^{T}, A_{21}^{T}, B_{1}^{T} , B_{2}^{T} , C_{1}^{T} , C_{2}^{T} } B_{2} B_{1}^{T} Σ^{−1} → −1 A_{21} − A_{12}^{T} Σ^{−1} Σ^{−1} B_{1} B_{2}^{T} → −1 Σ^{−1} A_{12} − A_{21}^{T} Σ^{−1} C_{1}^{T} C_{2} → −1 A_{12} − Σ^{−1} A_{21}^{T} C_{2}^{T} C_{1} Σ^{−1} → −1 A_{21} Σ^{−1} − A_{12}^{T} A_{21} B_{1} B_{1}^{T} Σ^{−1} → −1 A_{21} A_{11} + A_{12}^{T} A_{11}^{T} Σ^{−1} + B_{2} B_{1}^{T} A_{11}^{T} Σ^{−1} A_{21} C_{1}^{T} C_{1} Σ^{−1} → −1 A_{21} A_{11}^{T} + A_{12}^{T} A_{11} Σ^{−1} + B_{2} B_{1}^{T} A_{11} Σ^{−1} Σ^{−1} A_{11} B_{1} B_{2}^{T} → A_{11}^{T} A_{21}^{T} − Σ^{−1} A_{11} A_{12} + Σ^{−1} B_{1} B_{1}^{T} A_{21}^{T} Σ^{−1} A_{11} C_{1}^{T} C_{2} → A_{11}^{T} A_{12} − Σ^{−1} A_{11} A_{21}^{T} + Σ^{−1} B_{1} B_{1}^{T} A_{12} Σ^{−1} B_{1} B_{1}^{T} Σ^{−1} → −1 Σ^{−1} A_{11} − A_{11}^{T} Σ^{−1} Σ^{−1} A_{11}^{T} B_{1} B_{2}^{T} → A_{11} A_{21}^{T} − Σ^{−1} A_{11}^{T} A_{12} + Σ^{−1} C_{1}^{T} C_{1} A_{21}^{T} Σ^{−1} A_{11}^{T} C_{1}^{T} C_{2} → A_{11} A_{12} − Σ^{−1} A_{11}^{T} A_{21}^{T} + Σ^{−1} C_{1}^{T} C_{1} A_{12} Σ^{−1} C_{1}^{T} C_{1} Σ^{−1} → −1 A_{11} Σ^{−1} − Σ^{−1} A_{11}^{T} A_{12}^{T} B_{1} B_{1}^{T} Σ^{−1} → −1 A_{12}^{T} A_{11} + A_{21} A_{11}^{T} Σ^{−1} + C_{2}^{T} C_{1} A_{11}^{T} Σ^{−1} A_{12}^{T} C_{1}^{T} C_{1} Σ^{−1} → −1 A_{12}^{T} A_{11}^{T} + A_{21} A_{11} Σ^{−1} + C_{2}^{T} C_{1} A_{11} Σ^{−1} A_{21} A_{11} B_{1} B_{1}^{T} Σ^{−1} → −1 A_{21} A_{11} A_{11} − A_{12}^{T} A_{11}^{T} A_{11}^{T} Σ^{−1} + A_{21} B_{1} B_{1}^{T} A_{11}^{T} Σ^{−1} − B_{2} B_{1}^{T} A_{11}^{T} A_{11}^{T} Σ^{−1} A_{21} A_{11} C_{1}^{T} C_{1} Σ^{−1} → −1 A_{21} A_{11} A_{11}^{T} − A_{12}^{T} A_{11}^{T} A_{11} Σ^{−1} + A_{21} B_{1} B_{1}^{T} A_{11} Σ^{−1} − B_{2} B_{1}^{T} A_{11}^{T} A_{11} Σ^{−1} A_{21} A_{11}^{T} B_{1} B_{1}^{T} Σ^{−1} → −1 A_{21} A_{11}^{T} A_{11} − A_{12}^{T} A_{11} A_{11}^{T} Σ^{−1} + A_{21} C_{1}^{T} C_{1} A_{11}^{T} Σ^{−1} − B_{2} B_{1}^{T} A_{11} A_{11}^{T} Σ^{−1} A_{21} A_{11}^{T} C_{1}^{T} C_{1} Σ^{−1} → −1 A_{21} A_{11}^{T} A_{11}^{T} − A_{12}^{T} A_{11} A_{11} Σ^{−1} + A_{21} C_{1}^{T} C_{1} A_{11} Σ^{−1} − B_{2} B_{1}^{T} A_{11} A_{11} Σ^{−1} Σ^{−1} A_{11} A_{11} B_{1} B_{2}^{T} → −1 A_{11}^{T} A_{11}^{T} A_{21}^{T} − Σ^{−1} A_{11} A_{11} A_{12} + Σ^{−1} A_{11} B_{1} B_{1}^{T} A_{21}^{T} − Σ^{−1} B_{1} B_{1}^{T} A_{11}^{T} A_{21}^{T} Σ^{−1} A_{11} A_{11} C_{1}^{T} C_{2} → −1 A_{11}^{T} A_{11}^{T} A_{12} − Σ^{−1} A_{11} A_{11} A_{21}^{T} + Σ^{−1} A_{11} B_{1} B_{1}^{T} A_{12} − Σ^{−1} B_{1} B_{1}^{T} A_{11}^{T} A_{12} Σ^{−1} A_{11} B_{1} B_{1}^{T} Σ^{−1} → −1 Σ^{−1} A_{11} A_{11} + A_{11}^{T} A_{11}^{T} Σ^{−1} + Σ^{−1} B_{1} B_{1}^{T} A_{11}^{T} Σ^{−1} Σ^{−1} A_{11} A_{11}^{T} B_{1} B_{2}^{T} → −1 A_{11}^{T} A_{11} A_{21}^{T} − Σ^{−1} A_{11} A_{11}^{T} A_{12} + Σ^{−1} A_{11} C_{1}^{T} C_{1} A_{21}^{T} − Σ^{−1} B_{1} B_{1}^{T} A_{11} A_{21}^{T} Σ^{−1} A_{11} A_{11}^{T} C_{1}^{T} C_{2} → −1 A_{11}^{T} A_{11} A_{12} − Σ^{−1} A_{11} A_{11}^{T} A_{21}^{T} + Σ^{−1} A_{11} C_{1}^{T} C_{1} A_{12} − Σ^{−1} B_{1} B_{1}^{T} A_{11} A_{12} Σ^{−1} A_{11} C_{1}^{T} C_{1} Σ^{−1} → −1 Σ^{−1} A_{11} A_{11}^{T} + A_{11}^{T} A_{11} Σ^{−1} + Σ^{−1} B_{1} B_{1}^{T} A_{11} Σ^{−1} Σ^{−1} A_{11}^{T} A_{11} B_{1} B_{2}^{T} → −1 A_{11} A_{11}^{T} A_{21}^{T} − Σ^{−1} A_{11}^{T} A_{11} A_{12} + Σ^{−1} A_{11}^{T} B_{1} B_{1}^{T} A_{21}^{T} − Σ^{−1} C_{1}^{T} C_{1} A_{11}^{T} A_{21}^{T} Σ^{−1} A_{11}^{T} A_{11} C_{1}^{T} C_{2} → −1 A_{11} A_{11}^{T} A_{12} − Σ^{−1} A_{11}^{T} A_{11} A_{21}^{T} + Σ^{−1} A_{11}^{T} B_{1} B_{1}^{T} A_{12} − Σ^{−1} C_{1}^{T} C_{1} A_{11}^{T} A_{12} Σ^{−1} A_{11}^{T} B_{1} B_{1}^{T} Σ^{−1} → A_{11} A_{11}^{T} Σ^{−1} − Σ^{−1} A_{11}^{T} A_{11} + Σ^{−1} C_{1}^{T} C_{1} A_{11}^{T} Σ^{−1} Σ^{−1} A_{11}^{T} A_{11}^{T} B_{1} B_{2}^{T} → −1 A_{11} A_{11} A_{21}^{T} − Σ^{−1} A_{11}^{T} A_{11}^{T} A_{12} + Σ^{−1} A_{11}^{T} C_{1}^{T} C_{1} A_{21}^{T} − Σ^{−1} C_{1}^{T} C_{1} A_{11} A_{21}^{T} Σ^{−1} A_{11}^{T} A_{11}^{T} C_{1}^{T} C_{2} → −1 A_{11} A_{11} A_{12} − Σ^{−1} A_{11}^{T} A_{11}^{T} A_{21}^{T} + Σ^{−1} A_{11}^{T} C_{1}^{T} C_{1} A_{12} − Σ^{−1} C_{1}^{T} C_{1} A_{11} A_{12} Σ^{−1} A_{11}^{T} C_{1}^{T} C_{1} Σ^{−1} → A_{11} A_{11} Σ^{−1} − Σ^{−1} A_{11}^{T} A_{11}^{T} + Σ^{−1} C_{1}^{T} C_{1} A_{11} Σ^{−1} A_{12}^{T} A_{11} B_{1} B_{1}^{T} Σ^{−1} → −1 A_{12}^{T} A_{11} A_{11} − A_{21} A_{11}^{T} A_{11}^{T} Σ^{−1} + A_{12}^{T} B_{1} B_{1}^{T} A_{11}^{T} Σ^{−1} − C_{2}^{T} C_{1} A_{11}^{T} A_{11}^{T} Σ^{−1} A_{12}^{T} A_{11} C_{1}^{T} C_{1} Σ^{−1} → −1 A_{12}^{T} A_{11} A_{11}^{T} − A_{21} A_{11}^{T} A_{11} Σ^{−1} + A_{12}^{T} B_{1} B_{1}^{T} A_{11} Σ^{−1} − C_{2}^{T} C_{1} A_{11}^{T} A_{11} Σ^{−1} A_{12}^{T} A_{11}^{T} B_{1} B_{1}^{T} Σ^{−1} → −1 A_{12}^{T} A_{11}^{T} A_{11} − A_{21} A_{11} A_{11}^{T} Σ^{−1} + A_{12}^{T} C_{1}^{T} C_{1} A_{11}^{T} Σ^{−1} − C_{2}^{T} C_{1} A_{11} A_{11}^{T} Σ^{−1} A_{12}^{T} A_{11}^{T} C_{1}^{T} C_{1} Σ^{−1} → −1 A_{12}^{T} A_{11}^{T} A_{11}^{T} − A_{21} A_{11} A_{11} Σ^{−1} + A_{12}^{T} C_{1}^{T} C_{1} A_{11} Σ^{−1} − C_{2}^{T} C_{1} A_{11} A_{11} Σ^{−1}

The expressions with unknown variables {P_{12}^{T}} and knowns {A_{11}, C_{1}, A_{11}^{T}} C_{1} P_{12}^{T} → 0 P_{12}^{T} A_{11}^{T} → −1 A_{11} P_{12}^{T} C_{1} A_{11} P_{12}^{T} → 0 C_{1} A_{11} A_{11} P_{12}^{T} → 0 C_{1} A_{11} A_{11} A_{11} P_{12}^{T} → 0

The expressions with unknown variables {P_{11}, Σ^{−1}} and knowns {A_{11}, A_{12}, A_{21}, B_{1}, B_{2}, C_{1}, A_{11}^{T}, A_{12}^{T}, A_{21}^{T}, B_{1}^{T} , B_{2}^{T} , C_{1}^{T} , C_{2}^{T} } P_{11} Σ^{−1} A_{11} → A_{11} P_{11} Σ^{−1} P_{11} Σ^{−1} A_{12} → −1 B_{1} B_{2}^{T} − P_{11} A_{21}^{T} P_{11} Σ^{−1} B_{1} → B_{1} Σ^{−1} A_{11} P_{11} → −1 Σ^{−1} B_{1} B_{1}^{T} − A_{11}^{T} Σ^{−1} P_{11} A_{12}^{T} Σ^{−1} P_{11} → −1 A_{21} P_{11} − B_{2} B_{1}^{T} B_{1}^{T} Σ^{−1} P_{11} → B_{1}^{T} Σ^{−1} A_{11} A_{11} P_{11} → −1 Σ^{−1} A_{11} B_{1} B_{1}^{T} + Σ^{−1} B_{1} B_{1}^{T} A_{11}^{T} + A_{11}^{T} A_{11}^{T} Σ^{−1} P_{11} A_{12}^{T} A_{11}^{T} Σ^{−1} P_{11} → A_{21} A_{11} P_{11} + A_{21} B_{1} B_{1}^{T} − B_{2} B_{1}^{T} A_{11}^{T} B_{1}^{T} A_{11}^{T} Σ^{−1} P_{11} → B_{1}^{T} A_{11}^{T} A_{12}^{T} A_{11}^{T} A_{11}^{T} Σ^{−1} P_{11} → −1 A_{21} A_{11} A_{11} P_{11} − A_{21} A_{11} B_{1} B_{1}^{T} + A_{21} B_{1} B_{1}^{T} A_{11}^{T} − B_{2} B_{1}^{T} A_{11}^{T} A_{11}^{T} B_{1}^{T} A_{11}^{T} A_{11}^{T} Σ^{−1} P_{11} → B_{1}^{T} A_{11}^{T} A_{11}^{T} C_{2}^{T} C_{1} A_{11}^{T} Σ^{−1} P_{11} → A_{12}^{T} A_{11} P_{11} + A_{12}^{T} B_{1} B_{1}^{T} − A_{21} A_{11}^{T} Σ^{−1} P_{11}

{A_{21} C_{1}^{T} C_{1} A_{11}^{T} Σ^{−1} P_{11} → \atop A_{21} A_{11}^{T} A_{11} P_{11} + A_{21} A_{11}^{T} B_{1} B_{1}^{T} + A_{12}^{T} A_{11} A_{11}^{T} Σ^{−1} P_{11} + B_{2} B_{1}^{T} A_{11} A_{11}^{T} Σ^{−1} P_{11}}

Σ^{−1} C_{1}^{T} C_{1} A_{11}^{T} Σ^{−1} P_{11} → −1 A_{11} A_{11}^{T} Σ^{−1} P_{11} + Σ^{−1} A_{11}^{T} A_{11} P_{11} + Σ^{−1} A_{11}^{T} B_{1} B_{1}^{T}

{A_{12}^{T} C_{1}^{T} C_{1} A_{11}^{T} Σ^{−1} P_{11} → \atop A_{12}^{T} A_{11}^{T} A_{11} P_{11} + A_{12}^{T} A_{11}^{T} B_{1} B_{1}^{T} + A_{21} A_{11} A_{11}^{T} Σ^{−1} P_{11} + C_{2}^{T} C_{1} A_{11} A_{11}^{T} Σ^{−1} P_{11}}

{B_{1}^{T} Σ^{−1} A_{11} A_{11} A_{11} P_{11} → −1 B_{1}^{T} A_{11}^{T} A_{11}^{T} A_{11}^{T} − B_{1}^{T} Σ^{−1} A_{11} A_{11} B_{1} B_{1}^{T} + B_{1}^{T} Σ^{−1} A_{11} B_{1} B_{1}^{T} A_{11}^{T} − \atop B_{1}^{T} Σ^{−1} B_{1} B_{1}^{T} A_{11}^{T} A_{11}^{T}}

{C_{2}^{T} C_{1} A_{11}^{T} A_{11}^{T} Σ^{−1} P_{11} → \atop −1 A_{12}^{T} A_{11} A_{11} P_{11} − A_{12}^{T} A_{11} B_{1} B_{1}^{T} + A_{12}^{T} B_{1} B_{1}^{T} A_{11}^{T} − A_{21} A_{11}^{T} A_{11}^{T} Σ^{−1} P_{11}}

The expressions with unknown variables {P_{12}, P_{11}} and knowns {A_{12}, B_{1}, A_{21}^{T}, B_{2}^{T} } P_{12} A_{21}^{T} → A_{12} + B_{1} B_{2}^{T} + P_{11} A_{21}^{T}

The expressions with unknown variables {P_{12}, Σ^{−1}} and knowns {A_{11}, A_{21}, C_{1}, A_{11}^{T}, A_{12}^{T}, B_{1}^{T} , C_{2}^{T} } Σ^{−1} A_{11} P_{12} → −1 A_{11}^{T} Σ^{−1} P_{12} B_{1}^{T} Σ^{−1} P_{12} → 0 B_{1}^{T} A_{11}^{T} Σ^{−1} P_{12} → 0 B_{1}^{T} Σ^{−1} A_{11} A_{11} P_{12} → 0 C_{2}^{T} C_{1} A_{11}^{T} Σ^{−1} P_{12} → A_{12}^{T} A_{11} P_{12} − A_{21} A_{11}^{T} Σ^{−1} P_{12} B_{1}^{T} Σ^{−1} A_{11} A_{11} A_{11} P_{12} → 0

The expressions with unknown variables {P_{22}, Σ^{−1}} and knowns {A_{11}, C_{1}, A_{11}^{T}, C_{1}^{T} } C_{1} P_{22} → C_{1} Σ^{−1} P_{22} A_{11}^{T} → −1 A_{11} P_{22} + A_{11} Σ^{−1} + Σ^{−1} A_{11}^{T} P_{22} C_{1}^{T} → Σ^{−1} C_{1}^{T} C_{1} A_{11} P_{22} → C_{1} A_{11} Σ^{−1} C_{1} A_{11} A_{11} P_{22} → C_{1} A_{11} A_{11} Σ^{−1} C_{1} A_{11} A_{11} A_{11} P_{22} → C_{1} A_{11} A_{11} A_{11} Σ^{−1} The expressions with unknown variables {P_{11}^{−1}, P_{11}} and knowns {} P_{11} P_{11}^{−1} → 1 P_{11}^{−1} P_{11} → 1

The expressions with unknown variables {P_{11}^{−1}, Σ^{−1}} and knowns {A_{11}, B_{1}, A_{11}^{T}, B_{1}^{T} } P_{11}^{−1} A_{11} → Σ^{−1} A_{11} − A_{11}^{T} P_{11}^{−1} + A_{11}^{T} Σ^{−1} P_{11}^{−1} B_{1} → Σ^{−1} B_{1} B_{1}^{T} P_{11}^{−1} → B_{1}^{T} Σ^{−1} B_{1}^{T} A_{11}^{T} P_{11}^{−1} → B_{1}^{T} A_{11}^{T} Σ^{−1} B_{1}^{T} A_{11}^{T} A_{11}^{T} P_{11}^{−1} → B_{1}^{T} A_{11}^{T} A_{11}^{T} Σ^{−1}

The expressions with unknown variables {P_{22}^{−1}, P_{22}} and knowns {} P_{22} P_{22}^{−1} → 1 P_{22}^{−1} P_{22} → 1

The expressions with unknown variables {P_{22}^{−1}, Σ^{−1}} and knowns {A_{11}, A_{12}, A_{21}, B_{2}, C_{1}, C_{2}, A_{11}^{T}, A_{12}^{T}, A_{21}^{T}, B_{1}^{T} , C_{1}^{T} , C_{2}^{T} } A_{21} Σ^{−1} P_{22}^{−1} → −1 A_{12}^{T} P_{22}^{−1} − C_{2}^{T} C_{1} C_{1} Σ^{−1} P_{22}^{−1} → C_{1} P_{22}^{−1} Σ^{−1} A_{11}^{T} → A_{11}^{T} P_{22}^{−1} Σ^{−1} P_{22}^{−1} Σ^{−1} A_{21}^{T} → −1 P_{22}^{−1} A_{12} − C_{1}^{T} C_{2} P_{22}^{−1} Σ^{−1} C_{1}^{T} → C_{1}^{T} Σ^{−1} A_{11}^{T} P_{22}^{−1} → −1 A_{11} Σ^{−1} P_{22}^{−1} − Σ^{−1} C_{1}^{T} C_{1} A_{21} A_{11} Σ^{−1} P_{22}^{−1} → A_{12}^{T} A_{11}^{T} P_{22}^{−1} + A_{12}^{T} C_{1}^{T} C_{1} − C_{2}^{T} C_{1} A_{11} C_{1} A_{11} Σ^{−1} P_{22}^{−1} → C_{1} A_{11} B_{2} B_{1}^{T} A_{11} Σ^{−1} P_{22}^{−1} → A_{21} A_{11}^{T} P_{22}^{−1} + A_{21} C_{1}^{T} C_{1} − A_{12}^{T} A_{11} Σ^{−1} P_{22}^{−1} C_{1} A_{11} A_{11} Σ^{−1} P_{22}^{−1} → C_{1} A_{11} A_{11} C_{1} Σ^{−1} A_{11}^{T} A_{11}^{T} P_{22}^{−1} → C_{1} A_{11} A_{11} − C_{1} Σ^{−1} A_{11}^{T} C_{1}^{T} C_{1} + C_{1} Σ^{−1} C_{1}^{T} C_{1} A_{11}

The expressions with unknown variables {Σ^{−1}, Σ} and knowns {A_{11}, B_{1}, C_{1}, A_{11}^{T}, B_{1}^{T} , C_{1}^{T} } Σ Σ^{−1} → 1 Σ^{−1} Σ → 1 Σ^{−1} A_{11} Σ → −1 Σ^{−1} B_{1} B_{1}^{T} − A_{11}^{T} Σ^{−1} A_{11}^{T} Σ → −1 A_{11} − Σ^{−1} C_{1}^{T} C_{1} Σ^{−1} A_{11} A_{11} Σ → A_{11}^{T} A_{11}^{T} − Σ^{−1} A_{11} B_{1} B_{1}^{T} + Σ^{−1} B_{1} B_{1}^{T} A_{11}^{T} Σ^{−1} A_{11} A_{11}^{T} Σ → A_{11}^{T} A_{11} − Σ^{−1} A_{11} C_{1}^{T} C_{1} + Σ^{−1} B_{1} B_{1}^{T} A_{11}

Σ^{−1} A_{11}^{T} A_{11} Σ → A_{11} A_{11}^{T} − Σ^{−1} A_{11}^{T} B_{1} B_{1}^{T} + Σ^{−1} C_{1}^{T} C_{1} A_{11}^{T}

Σ^{−1} A_{11}^{T} A_{11}^{T} Σ → A_{11} A_{11} − Σ^{−1} A_{11}^{T} C_{1}^{T} C_{1} + Σ^{−1} C_{1}^{T} C_{1} A_{11}

Σ^{−1} A_{11} A_{11} A_{11} Σ → −1 A_{11}^{T} A_{11}^{T} A_{11}^{T}−Σ^{−1} A_{11} A_{11} B_{1} B_{1}^{T} +Σ^{−1} A_{11} B_{1} B_{1}^{T} A_{11}^{T}−Σ^{−1} B_{1} B_{1}^{T} A_{11}^{T} A_{11}^{T}

Σ^{−1} A_{11} A_{11} A_{11}^{T} Σ → −1 A_{11}^{T} A_{11}^{T} A_{11}−Σ^{−1} A_{11} A_{11} C_{1}^{T} C_{1}+Σ^{−1} A_{11} B_{1} B_{1}^{T} A_{11}−Σ^{−1} B_{1} B_{1}^{T} A_{11}^{T} A_{11}

Σ^{−1} A_{11} A_{11}^{T} A_{11} Σ → −1 A_{11}^{T} A_{11} A_{11}^{T}−Σ^{−1} A_{11} A_{11}^{T} B_{1} B_{1}^{T} +Σ^{−1} A_{11} C_{1}^{T} C_{1} A_{11}^{T}−Σ^{−1} B_{1} B_{1}^{T} A_{11} A_{11}^{T}

Σ^{−1} A_{11} A_{11}^{T} A_{11}^{T} Σ → −1 A_{11}^{T} A_{11} A_{11}−Σ^{−1} A_{11} A_{11}^{T} C_{1}^{T} C_{1}+Σ^{−1} A_{11} C_{1}^{T} C_{1} A_{11}−Σ^{−1} B_{1} B_{1}^{T} A_{11} A_{11}

Σ^{−1} A_{11}^{T} A_{11} A_{11} Σ → −1 A_{11} A_{11}^{T} A_{11}^{T}−Σ^{−1} A_{11}^{T} A_{11} B_{1} B_{1}^{T} +Σ^{−1} A_{11}^{T} B_{1} B_{1}^{T} A_{11}^{T}−Σ^{−1} C_{1}^{T} C_{1} A_{11}^{T} A_{11}^{T}

Σ^{−1} A_{11}^{T} A_{11} A_{11}^{T} Σ → −1 A_{11} A_{11}^{T} A_{11}−Σ^{−1} A_{11}^{T} A_{11} C_{1}^{T} C_{1}+Σ^{−1} A_{11}^{T} B_{1} B_{1}^{T} A_{11}−Σ^{−1} C_{1}^{T} C_{1} A_{11}^{T} A_{11}

Σ^{−1} A_{11}^{T} A_{11}^{T} A_{11} Σ → −1 A_{11} A_{11} A_{11}^{T}−Σ^{−1} A_{11}^{T} A_{11}^{T} B_{1} B_{1}^{T} +Σ^{−1} A_{11}^{T} C_{1}^{T} C_{1} A_{11}^{T}−Σ^{−1} C_{1}^{T} C_{1} A_{11} A_{11}^{T}

Σ^{−1} A_{11}^{T} A_{11}^{T} A_{11}^{T} Σ → −1 A_{11} A_{11} A_{11}−Σ^{−1} A_{11}^{T} A_{11}^{T} C_{1}^{T} C_{1}+Σ^{−1} A_{11}^{T} C_{1}^{T} C_{1} A_{11}−Σ^{−1} C_{1}^{T} C_{1} A_{11} A_{11}

The expressions with unknown variables {P_{12}^{T}, P_{11}} and knowns {A_{11}, A_{21}, B_{1}, B_{2}, A_{11}^{T}, A_{12}^{T}, B_{1}^{T} } A_{21} P_{12}^{T} → A_{21} P_{11} + B_{2} B_{1}^{T} + A_{12}^{T}

A_{21} A_{11} P_{12}^{T} → −1 A_{12}^{T} A_{11}^{T} + A_{21} A_{11} P_{11} + A_{21} B_{1} B_{1}^{T} − B_{2} B_{1}^{T} A_{11}^{T}

A_{21} A_{11} A_{11} P_{12}^{T} → A_{12}^{T} A_{11}^{T} A_{11}^{T} +A_{21} A_{11} A_{11} P_{11} +A_{21} A_{11} B_{1} B_{1}^{T} −A_{21} B_{1} B_{1}^{T} A_{11}^{T} +B_{2} B_{1}^{T} A_{11}^{T} A_{11}^{T}

{A_{21} A_{11} A_{11} A_{11} P_{12}^{T} → −1 A_{12}^{T} A_{11}^{T} A_{11}^{T} A_{11}^{T} + A_{21} A_{11} A_{11} A_{11} P_{11} + A_{21} A_{11} A_{11} B_{1} B_{1}^{T} − \atop A_{21} A_{11} B_{1} B_{1}^{T} A_{11}^{T} + A_{21} B_{1} B_{1}^{T} A_{11}^{T} A_{11}^{T} − B_{2} B_{1}^{T} A_{11}^{T} A_{11}^{T} A_{11}^{T}}

The expressions with unknown variables {P_{12}^{T}, Σ^{−1}} and knowns {A_{11}, B_{1}} P_{12}^{T} Σ^{−1} A_{11} → A_{11} P_{12}^{T} Σ^{−1}

P_{12}^{T} Σ^{−1} B_{1} → 0

The expressions with unknown variables {P_{12}, P_{22}, Σ^{−1}} and knowns {A_{11}, A_{21}, A_{11}^{T}, A_{12}^{T}}

A_{21} P_{12} → A_{21} P_{22} − A_{21} Σ^{−1}

A_{21} A_{11} P_{12} → A_{21} A_{11} P_{22} − A_{21} A_{11} Σ^{−1}

A_{12}^{T} Σ^{−1} P_{12} → −1 A_{21} P_{22} + A_{21} Σ^{−1}

A_{21} A_{11} A_{11} P_{12} → A_{21} A_{11} A_{11} P_{22} − A_{21} A_{11} A_{11} Σ^{−1}

A_{12}^{T} A_{11}^{T} Σ^{−1} P_{12} → A_{21} A_{11} P_{22} − A_{21} A_{11} Σ^{−1}

A_{21} A_{11} A_{11} A_{11} P_{12} → A_{21} A_{11} A_{11} A_{11} P_{22} − A_{21} A_{11} A_{11} A_{11} Σ^{−1}

A_{12}^{T} Σ^{−1} A_{11} A_{11} P_{12} → −1 A_{21} A_{11} A_{11} P_{22} + A_{21} A_{11} A_{11} Σ^{−1}

The expressions with unknown variables {P_{22}, P_{11}, Σ^{−1}} and knowns {A_{11}, A_{21}, B_{1}, B_{2}, A_{11}^{T}, A_{12}^{T}, A_{21}^{T}, B_{1}^{T} } A_{21} P_{22} A_{21}^{T} → A_{12}^{T} A_{21}^{T} + A_{21} P_{11} A_{21}^{T} + A_{21} Σ^{−1} A_{21}^{T} + B_{2} B_{1}^{T} A_{21}^{T}

{A_{21} A_{11} P_{22} A_{21}^{T} → −1 A_{12}^{T} A_{11}^{T} A_{21}^{T} + A_{21} A_{11} P_{11} A_{21}^{T} + A_{21} A_{11} Σ^{−1} A_{21}^{T} + A_{21} B_{1} B_{1}^{T} A_{21}^{T} − \atop B_{2} B_{1}^{T} A_{11}^{T} A_{21}^{T}}

{A_{21} A_{11} A_{11} P_{22} A_{21}^{T} → A_{12}^{T} A_{11}^{T} A_{11}^{T} A_{21}^{T} + A_{21} A_{11} A_{11} P_{11} A_{21}^{T} + A_{21} A_{11} A_{11} Σ^{−1} A_{21}^{T} + \atop A_{21} A_{11} B_{1} B_{1}^{T} A_{21}^{T} − A_{21} B_{1} B_{1}^{T} A_{11}^{T} A_{21}^{T} + B_{2} B_{1}^{T} A_{11}^{T} A_{11}^{T} A_{21}^{T}}

The expressions with unknown variables {P_{12}^{T}, P_{22}, Σ^{−1}} and knowns {A_{12}, A_{21}^{T}} P_{12}^{T} A_{21}^{T} → P_{22} A_{21}^{T} − Σ^{−1} A_{21}^{T}

P_{12}^{T} Σ^{−1} A_{12} → −1 P_{22} A_{21}^{T} + Σ^{−1} A_{21}^{T}