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9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 | import OAI.Analysis.HyperbolicCones.RawBlocks
import OAI.Analysis.HyperbolicCones.CongruenceMaps
namespace OAI
/-! Fixed positive-definite diagonal congruence makes both positive maps unital. -/
noncomputable section
open scoped Matrix.Norms.L2Operator MatrixOrder
open Matrix Set
namespace Paper256
noncomputable def RawBlockPencil.normalize {K : Set Ambient} (P : RawBlockPencil K) :
BlockPencil K := by
let S := inverseSquareRoot (P.D 1)
let T := inverseSquareRoot (P.E 1)
refine {
a := P.a
c := P.c
a_pos := P.a_pos
c_pos := P.c_pos
D := (congruenceSym S).comp P.D
E := (congruenceSym T).comp P.E
F := (congruenceSym S).comp P.F
C := (congruenceRect S T).comp P.C
A := (congruenceSym S).comp P.A
B := (congruenceRect S T).comp P.B
G := (congruenceSym T).comp P.G
D_positive := fun X hX => congruenceSym_positive S (P.D X) (P.D_positive X hX)
E_positive := fun Z hZ => congruenceSym_positive T (P.E Z) (P.E_positive Z hZ)
D_unital := congruenceSym_inverseSquareRoot_self (P.D 1) P.D_identity_posDef
E_unital := congruenceSym_inverseSquareRoot_self (P.E 1) P.E_identity_posDef
represents := ?_ }
intro X Z y
rw [P.represents]
have h := block_congruence_posSemidef_iff S T
(inverseSquareRoot_isUnit (P.D 1) P.D_identity_posDef)
(inverseSquareRoot_isUnit (P.E 1) P.E_identity_posDef)
((P.D X : Mat P.a ℝ) + (P.F Z : Mat P.a ℝ) + (P.A y : Mat P.a ℝ))
(P.C Z + P.B y) ((P.E Z : Mat P.c ℝ) + (P.G y : Mat P.c ℝ))
simpa only [LinearMap.comp_apply, congruenceSym_val, congruenceRect_apply,
Matrix.mul_add, Matrix.add_mul] using h.symm
end Paper256
end
end OAI
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