512 B The inverse of a multivariate homogeneous polynomial
The elaborate algebraic manipulation becomes more clear when we consider Examples B.3 and B.4.
The first example illustrates our intention to invert a vector-valued bivariate homogeneous polynomial
of degree n = 2, namely y(x) = A 11 x + A 12 x ⊗ x → x(y) = B 11 y + B 12 y ⊗ y. The Kronecker–Zehfuss
product of column arrays is explicitly given. The GKS algorithm determines the set of matrices
{B 11 , B 12 } from the two given matrices A 11 and A 12 . In contrast, the second example introduces the
problem of inversion of a vector-valued bivariate homogeneous polynomial of degree n = 3, namely
y(x) = A 11 x + A 12 x ⊗ x + A 13 x ⊗ x ⊗ x → x(y) = B 11 y + B 12 y ⊗ y + B 13 y ⊗ y ⊗ y. An explicit
representation of the Kronecker–Zehfuss product of double and triple column arrays is presented.
Again, the GKS algorithm determines the set of matrices {B 11 , B 12 , B 13 } from the three given matrices
A 11 , A 12 , and A 13 .
Example B.3 (Inversion of a bivariate homogeneous polynomial of degeree n = 2).
Assume the bivariate homogeneous polynomial y(x) = A 11 x + A 12 x ⊗ x to be given and find the
inverse bivariate homogeneous polynomial x(y) = B 11 y + B 12 y ⊗ y by the GKS algorithm.
Basic equations:
y(x) =
y 1
y 2
= A 11
x 1
x 2
+ A 12
x 1
x 2
⊗
x 1
x 2
(B.34)
or
y 1 = a
11
11 x 1 + a
12
11 x 2 + a
11
12 x
2
1 +
a
12
12 + a
13
12
x 1 x 2 + a
14
12 x
2
2 ,
y 2 = a
21
11 x 1 + a
22
11 x 2 + a
21
12 x
2
1 +
a
22
12 + a
23
12
x 1 x 2 + a
24
12 x
2
2 ,
(B.35)
x 1
x 2
⊗
x 1
x 2
=
x 1
x 2
[2]
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
x
2
1
x 1 x 2
x 2 x 1
x
2
2
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(B.36)
(Kronecker–Zehfuss product);
x(y) =
x 1
x 2
= B 11
y 1
y 2
+ B 12
y 1
y 2
⊗
y 1
y 2
(B.37)
or
x 1 = b
11
11 y 1 + b
12
11 y 2 + b
11
12 y
2
1 +
b
12
12 + b
13
12
y 1 y 2 + b
14
12 y
2
2 ,
x 2 = b
21
11 y 1 + b
22
11 y 2 + b
21
12 y
2
1 +
b
22
12 + b
23
12
y 1 y 2 + b
24
12 y
2
2 ,
(B.38)
y 1
y 2
⊗
y 1
y 2
=
y 1
y 2
[2]
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
y
2
1
y 1 y 2
y 2 y 1
y
2
2
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(B.39)
(Kronecker–Zehfuss product).