Cholesky Decomposition Index   <<   >>
  

┌───┬───┐     ┌──────┬───────┐   ┌────────┬─────────┐
│ A │ B │     │  L0  │   0   │   │  +⍉L0  │+⍉ T x L0│
├───┼───┤  =  ├──────┼───────┤ x ├────────┼─────────┤
│+⍉B│ C │     │T x L0│   L1  │   │    0   │   +⍉L1  │
└───┴───┘     └──────┴───────┘   └────────┴─────────┘
✔ (a)   A  = L0 x +⍉L0
✔ (b)   B  = L0 x +⍉ T x L0
✔ (c)   (+⍉B)  = (T x L0) x +⍉L0
  (d)   C  = ((T x L0) x (+⍉ T x L0)) + (L1 x +⍉L1)

((T x L0) x (+⍉ T x L0)) + (L1 x +⍉L1)    
((T x L0) x (+⍉ T x L0)) + C - T x B L1←∇ C - T x B
((T x L0) x (+⍉L0) x +⍉T) + C - T x B +⍉ X x Y ←→ (+⍉Y) x +⍉X
(T x L0 x (+⍉L0) x +⍉T) + C - T x B associativity of x
(T x A x +⍉T) + C - T x B (a)
(T x A x +⍉ (+⍉B) x ⌹A) + C - T x B T←(+⍉B) x ⌹A
(T x A x (+⍉⌹A) x B) + C - T x B +⍉ X x Y ←→ (+⍉Y) x +⍉X
(T x A x (⌹A) x B) + C - T x B A and hence ⌹A are Hermitian
(T x I x B) + C - T x B matrix inverse
(T x B) + C - T x B identity matrix
C