Essays/Triangular Matrix Inverse
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A matrix is triangular if it satisfies one of the following propositions:
upperQ=: 0&= >: >/&i.@$ lowerQ=: 0&= >: </&i.@$
A recursive method for inverting a square triangular matrix reveals itself by considering it as a 2-by-2 matrix of matrices and substituting matrix operations for scalar ones in the inversion of a 2-by-2 triangular matrix of scalars.
uinv=: 3 : 0
if. 1>:#y do. %y return. end.
m =. >.-:#y
n =. <.-:#y
ai=. uinv (m,m){.y
di=. uinv (m,m)}.y
b =. (m,-n){.y
x =. - ai +/ .* b +/ .* di
(ai,.x) , (-m+n){."1 di
)
linv=: 3 : 0
if. 1>:#y do. %y return. end.
m =. >.-:#y
n =. <.-:#y
ai=. linv (m,m){.y
di=. linv (m,m)}.y
c =. ((-n),m){.y
y =. - di +/ .* c +/ .* ai
((m+n){."1 ai) , (y,.di)
)
Since uinv -: linv&.|: and linv -: uinv&.|: , one of the verbs can be defined simply in terms of the other.
Some examples using Pascal's triangle and Stirling numbers:
] A=: !/~ i.7x 1 1 1 1 1 1 1 0 1 2 3 4 5 6 0 0 1 3 6 10 15 0 0 0 1 4 10 20 0 0 0 0 1 5 15 0 0 0 0 0 1 6 0 0 0 0 0 0 1 uinv A 1 _1 1 _1 1 _1 1 0 1 _2 3 _4 5 _6 0 0 1 _3 6 _10 15 0 0 0 1 _4 10 _20 0 0 0 0 1 _5 15 0 0 0 0 0 1 _6 0 0 0 0 0 0 1 A +/ .* uinv A 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 s1=: 1: ` ([ ((0:,]) + <:@[ * ],0:) $:@<:) @. * s2=: 1: ` (i.@>: ((0:,]) + [ * ],0:) $:@<:) @. * s1"0 i.7x 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 2 3 1 0 0 0 0 6 11 6 1 0 0 0 24 50 35 10 1 0 0 120 274 225 85 15 1 s2"0 i.7x 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 3 1 0 0 0 0 1 7 6 1 0 0 0 1 15 25 10 1 0 0 1 31 90 65 15 1 | linv s1"0 i.7x 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 3 1 0 0 0 0 1 7 6 1 0 0 0 1 15 25 10 1 0 0 1 31 90 65 15 1
See also
- Cholesky Decomposition
- LU Decomposition
- QR Decomposition
- Matrix Inverse
- Triangular Matrix Inverse
- Determinant
- Minors
- Hilbert Matrix
- Block Matrix Inverse
- Kronecker Product
Contributed by Roger Hui.