Tests, Derivations, Proofs

Cholesky Decomposition


┌───┬───┐     ┌──────┬───────┐   ┌────────┬─────────┐
│ 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)`


┌──────┬───────┐
│  L0  │   0   │
├──────┼───────┤
│T x L0│   L1  │
└──────┴───────┘

Finally, L is lower triangular if L0 and L1 are lower triangular,
and they are by induction (recursion).