518 B The inverse of a multivariate homogeneous polynomial
Box B.4 (Inversion of a multivariate homogeneous polynomial of degree n: the upper triangular matrix
A := A ).
Recurrence relation:
A mn =
n−(m−1)
X
i=1
A 1i ⊗ A m−1n−i ∀ m ≤ n .
(B.66)
Inversion relations (A 11 , A 1n given):
A 22 = A
[2]
11 , A 2n =
n−1 X
i=1
A 1i ⊗ A 1n−i ,
A 33 = A
[3]
11 , A 3n =
n−2 X
i=1
A 1i ⊗
n−i−1 X
j=1
A 1j ⊗ A 1n−i−j ,
A 44 = A
[4]
11 , A 4n =
n−3 X
i=1
A 1i ⊗
n−i−2 X
j=1
A 1j ⊗
n−i−j−1
X
k=1
A 1k ⊗ A 1n−i−j−k ,
A 55 = A
[5]
11 , A 5n =
n−4 X
i=1
A 1i ⊗
n−i−3 X
j=1
A 1j ⊗
n−i−j−2
X
k=1
A 1k ⊗
n−i−j−k−1
X
l=1
A 1l ⊗ A 1n−i−j−k−l ,
. . .
A mn =
n−(m−1)
X
k 1 =1
A 1k 1 ⊗
n−k 1 −(m−2)
X
k 2 =1
A 1k 2 ⊗
n−k 1 −k 2 −(m−3)
X
k 3 =1
A 1k 3 ⊗ · · · ⊗
⊗ · · · ⊗
n−k 1 −k 2 −···−k m−2
X
k m−1 =1
A 1k m−1 ⊗ A 1n−k 1 −k 2 −···−k m−2 −k m−1 .
(B.67)
Box B.5 (Inversion of a multivariate homogeneous polynomial of degree n: 1st row of the triangular matrix
B := B ).
Recurrence relation:
B 1n = −
n−1 X
i=1
B 1i A in
!
A
−1
nn ∀ n ≥ 2 , subject to A
−1
nn =
“
A
[n]
11
” −1
=
`
A
−1
11
´ [n] .
(B.68)
Inversion relations (A 11 , A 1n given):
B 11 = +A
−1
11 , B 12 = −A
−1
11 A 12 A
−1
22 , B 13 = +A
−1
11
ˆ
A 12 A
−1
22 A 23 − A 13
˜
A
−1
33 ,
B 14 = +A
−1
11
ˆ
A 12 A
−1
22
`
A 24 − A 23 A
−1
33 A 34
´
+ A 13 A
−1
33 A 34 − A 14
˜
A
−1
44 ,
B 15 = +A
−1
11
ˆ
A 12 A
−1
22
`
A 25 − A 24 A
−1
44 A 45 − A 23 A
−1
33
`
A 35 − A 34 A
−1
44 A 45
´´
+
+A 13 A
−1
33
`
A 35 − A 34 A
−1
44 A 45
´
+ A 14 A
−1
44 A 45 − A 15
˜
A
−1
55 ,
. . .
B 1n = +A
−1
11
»
− A 1n +
n−1 X
i=2
A 1i A
−1
ii A in −
n−2 X
i=2
A 1i A
−1
ii A in−1 A
−1
n−1n−1 A n−1n −
−
n−3 X
i=2
A 1i A
−1
ii A in−2 A
−1
n−2n−2
`
A n−2n − A n−2n−1 A
−1
n−1n−1 A n−1n
´ − · · ·
–
∀ n ≥ 2 .
(B.69)