Input = A P inv[A] A == A P inv[A] A P inv[A] == P inv[A] A^{T} (P inv[A])^{T} == P inv[A] A (P inv[A])^{T} A^{T} == A P inv[A] (P inv[A])^{T} == P inv[A^{T}] A^{T} A P inv[A] == A^{T} A P inv[A] P inv[A^{T}] == P inv[A^{T}] B P inv[B] B == B P inv[B] B P inv[B] == P inv[B] B^{T} (P inv[B])^{T} == P inv[B] B (P inv[B])^{T} B^{T} == B P inv[B] (P inv[B])^{T} == P inv[B^{T}] B^{T} B P inv[B] == B^{T} B P inv[B] P inv[B^{T}] == P inv[B^{T}] (A P inv[A]−P inv[B] B A P inv[A]) P inv[A P inv[A]−P inv[B] B A P inv[A]] (A P inv[A]−P inv[B] B A P inv[A]) == A P inv[A] − P inv[B] B A P inv[A] P inv[A P inv[A]−P inv[B] B A P inv[A]] (A P inv[A]−P inv[B] B A P inv[A]) P inv[A P inv[A]−P inv[B] B A P inv[A]] == P inv[A P inv[A] − P inv[B] B A P inv[A]] ((P inv[A])^{T} A^{T} −(P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}) (P inv[A P inv[A] − P inv[B] B A P inv[A]])^{T} == P inv[A P inv[A] − P inv[B] B A P inv[A]] (A P inv[A] − P inv[B] B A P inv[A]) (P inv[A P inv[A] − P inv[B] B A P inv[A]])^{T} ((P inv[A])^{T} A^{T} −(P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}) == (A P inv[A] − P inv[B] B A P inv[A]) P inv[A P inv[A] − P inv[B] B A P inv[A]] (P inv[A P inv[A] − P inv[B] B A P inv[A]])^{T} == P inv[(P inv[A])^{T} A^{T} −(P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] ((P inv[A])^{T} A^{T} −(P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}) (A P inv[A]−P inv[B] B A P inv[A]) P inv[A P inv[A]− P inv[B] B A P inv[A]] == (P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T} (A P inv[A]−P inv[B] B A P inv[A]) P inv[A P inv[A]−P inv[B] B A P inv[A]] P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] == P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] File Name = APinvAMinusPinvBBAPinvA NCMakeGB Iterations = 6 NCMakeGB on Digested Iterations = 0 SmallBasis Iterations = 0 SmallBasis on Knowns Iterations = 0 Deselect→ {} UserSelect→ {} UserUnknowns→ {} NCShortFormulas→-1 RR→False RRByCat→False SB→False SBByCat→False DegreeCap→-1 DegreeSumCap→-1 DegreeCapSB→-1 DegreeSumCapSB→-1 NCCV→False THE ORDER IS NOW THE FOLLOWING: A < B < A^{T} < B^{T} < P inv[A] < P inv[B] < P inv[A^{T}] < P inv[B^{T}] < (P inv[A])^{T} < (P inv[B])^{T} < P inv[A P inv[A]−P inv[B] B A P inv[A]] < P inv[(P inv[A])^{T} A^{T} −(P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] <

(P inv[A P inv[A] − P inv[B] B A P inv[A]])^{T}

YOUR SESSION HAS DIGESTED THE FOLLOWING RELATIONS THE FOLLOWING VARIABLES HAVE BEEN SOLVED FOR: {(P inv[A])^{T} , (P inv[B])^{T} , (P inv[A P inv[A] − P inv[B] B A P inv[A]])^{T} } The corresponding rules are the following: (P inv[A])^{T} → P inv[A^{T}] (P inv[B])^{T} → P inv[B^{T}]

{(P inv[A P inv[A] − P inv[B] B A P inv[A]])^{T}  →  P inv[(P inv[A])^{T} A^{T}  − \atop (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}]}

The expressions with unknown variables {} and knowns {A, B, P inv[A], P inv[B], P inv[A P inv[A]−P inv[B] B A P inv[A]], P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}], P inv[A^{T}], P inv[B^{T}], A^{T} , B^{T} } B P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] → 0 P inv[A] A → A^{T} P inv[A^{T}] P inv[B] B → B^{T} P inv[B^{T}] P inv[A P inv[A] − P inv[B] B A P inv[A]] P inv[B] → 0

{P inv[A P inv[A] − P inv[B] B A P inv[A]] P inv[A P inv[A] − P inv[B] B A P inv[A]] → \atop P inv[A P inv[A] − P inv[B] B A P inv[A]]}

P inv[A P inv[A] − P inv[B] B A P inv[A]] B^{T} → 0

{P inv[(P inv[A])^{T} A^{T}  −  (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] P inv[(P inv[A])^{T} A^{T}  − \atop (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] → P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}]}

P inv[A^{T}] A^{T} → A P inv[A] P inv[B^{T}] P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] → 0 P inv[B^{T}] B^{T} → B P inv[B] A P inv[A] P inv[A P inv[A] − P inv[B] B A P inv[A]] → P inv[A P inv[A] − P inv[B] B A P inv[A]] A P inv[A] P inv[A^{T}] → P inv[A^{T}] A A^{T} P inv[A^{T}] → A B P inv[B] P inv[B^{T}] → P inv[B^{T}] B B^{T} P inv[B^{T}] → B

{P inv[A] P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] A → P inv[A] P inv[A P inv[A] − \atop P inv[B] B A P inv[A]] A}

{P inv[A] P inv[(P inv[A])^{T} A^{T}  −  (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] P inv[A P inv[A]  − \atop P inv[B] B A P inv[A]] → P inv[A] P inv[A P inv[A] − P inv[B] B A P inv[A]]}
{P inv[A] P inv[(P inv[A])^{T} A^{T}  −  (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] P inv[A^{T}]  → \atop P inv[A] P inv[A P inv[A] − P inv[B] B A P inv[A]] P inv[A^{T}]}

{P inv[A P inv[A] − P inv[B] B A P inv[A]] A P inv[A] → A P inv[A] P inv[(P inv[A])^{T} A^{T} − \atop (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}]}

{P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] A P inv[A] → P inv[(P inv[A])^{T} A^{T} − \atop (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}]}

{P inv[(P inv[A])^{T} A^{T}  −  (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] P inv[A P inv[A]  − \atop P inv[B] B A P inv[A]] A → P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] A}
{P inv[(P inv[A])^{T} A^{T}  −  (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] P inv[A P inv[A]  − \atop P inv[B] B A P inv[A]] P inv[A^{T}] → P inv[(P inv[A])^{T} A^{T} −(P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] P inv[A^{T}]}

P inv[(P inv[A])^{T} A^{T}  −  (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] B^{T} P inv[B^{T}]  →

B^{T} P inv[B^{T}] P inv[A P inv[A] − P inv[B] B A P inv[A]] − P inv[A P inv[A] − P inv[B] B A P inv[A]] + P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}]

A^{T} A P inv[A] → A^{T}

{A^{T} P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] A → A^{T} P inv[A P inv[A] − \atop P inv[B] B A P inv[A]] A}

{A^{T} P inv[(P inv[A])^{T} A^{T}  −  (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] P inv[A P inv[A]  − \atop P inv[B] B A P inv[A]] → A^{T} P inv[A P inv[A] − P inv[B] B A P inv[A]]}

{A^{T} P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] P inv[A^{T}] → A^{T} P inv[A P inv[A] − \atop P inv[B] B A P inv[A]] P inv[A^{T}]}

A^{T} P inv[A^{T}] P inv[A] → P inv[A]

B^{T} B P inv[B] → B^{T}

B^{T} P inv[B^{T}] P inv[B] → P inv[B]

{P inv[A] B^{T} P inv[B^{T}] P inv[A P inv[A] − P inv[B] B A P inv[A]] → P inv[A] P inv[A P inv[A] − \atop P inv[B] B A P inv[A]] + P inv[A] B^{T} P inv[B^{T}] − P inv[A]}

{A^{T} B^{T} P inv[B^{T}] P inv[A P inv[A] − P inv[B] B A P inv[A]]  →  A^{T} P inv[A P inv[A] − \atop P inv[B] B A P inv[A]] + A^{T} B^{T} P inv[B^{T}] − A^{T}}

{B^{T} P inv[B^{T}] P inv[A P inv[A] − P inv[B] B A P inv[A]] A → −1 A + P inv[A P inv[A] − \atop P inv[B] B A P inv[A]] A + B^{T} P inv[B^{T}] A}

B^{T} P inv[B^{T}] P inv[A P inv[A]  −  P inv[B] B A P inv[A]] P inv[(P inv[A])^{T} A^{T}  −

(P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] → P inv[A P inv[A]−P inv[B] B A P inv[A]] P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] − P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}]

{B^{T} P inv[B^{T}] P inv[A P inv[A] − P inv[B] B A P inv[A]] P inv[A^{T}]  →  P inv[A P inv[A] − \atop P inv[B] B A P inv[A]] P inv[A^{T}] + B^{T} P inv[B^{T}] P inv[A^{T}] − P inv[A^{T}]}

{B^{T} P inv[B^{T}] A P inv[A] P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] → −1 A P inv[A] + \atop A P inv[A] P inv[(P inv[A])^{T} A^{T} − (P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}] + B^{T} P inv[B^{T}] A P inv[A]}

USER CREATIONS APPEAR BELOW

SOME RELATIONS WHICH APPEAR BELOW MAY BE UNDIGESTED

THE FOLLOWING VARIABLES HAVE NOT BEEN SOLVED FOR: {A, B, P inv[A], P inv[B], P inv[A P inv[A]−P inv[B] B A P inv[A]], P inv[(P inv[A])^{T} A^{T} −(P inv[A])^{T} A^{T} B^{T} (P inv[B])^{T}], P inv[A^{T}], P inv[B^{T}], A^{T} , B^{T} }

The time for preliminaries was 0:00:00 The time for NCMakeGB 1 was 0:00:00 The time for Remove Redundant 1 was 0:00:00 The time for the main NCMakeGB was 0:00:04 The time for Remove Redundant 2 was 0:00:00 The time for reducing unknowns was 0:00:00 The time for clean up basis was 0:00:00 The time for SmallBasis was 0:00:00 The time for CreateCategories was 0:00:04 The time for NCCV was 0:00:00 The time for RegularOutput was 0:00:00 The time for everything so far was 0:00:08

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