The Inverse of a Multivariate Homogeneous Polynomial - Map Projections: Cartographic Information Systems - page 689

Backward substitution:

⎡ ⎤ ⎡ ⎤

B 11 B 12 B 13
0B 22 B 23
00B 33

A 11 A 12 A 13
0A 22 A 23
00A 33

⎣ ⎦ ⎣ ⎦ =I 14 ;

B =I 14 ⇔

(B.57)

A

(i)B 11 A 11 =I 2
⇒
B 11 =A − 1

11 ;
(ii)B 12 A 12 +B 12 A 22 =0
⇒

11 A 12 A − 1[2]

B 11 A 12 A − 1 A − 1

B 12 = −

22 =

−

22 ;

(B.58)

(iii)B 11 A 13 +B 12 A 23 +B 13 A 33 =0
⇒

(B 11 A 13 +B 12 A 23 )A − 1 B 13 = −

33 =

11 A 13 A − 1[3] 11 A 12 A − 1[2] 11 (A 11 ⊗ A 12 +A 12 ⊗ A 11 )A − 1[3]

= − A − 1 +A − 1

11 11

=A − 1 A 13 +A 12 (A − 1 A − 1 A 11 )](A − 1 A − 1

11 A − 1 11 [

− ⊗

11 )(A 11

⊗ A 12 +A 12 ⊗ ⊗

11 ) . 11 11

First, we have used ( A⊗B ) − 1 = A − 1

⊗B − 1 for two invertible square matrices A and B , secondly
we have used the standard solution of a system of upper triangular matrix equations. For the
inverse polynomial representation, only the elements of the first row of the matrix B :=

B are

of interest. An explicit write-up is
x 1
x 2

=A − 1 y 1

y 2

11 ) y 1

[2]

A − 1

11 A 12 (A − 1

A − 1

− ⊗ −

11 11

y 2

A − 1 A 12 (A − 1 A − 1 −

11 [A 13 −

⊗

11 )(A 11 ⊗

A 12 +A 12 ⊗ A 11 )]

11

11 ) y 1

[3]

(A − 1 ⊗ A − 1 ⊗ A − 1

. (B.59)

11 11

y 2

End of Example.

B-3 Inversion of a Multivariate Homogeneous Polynomial of Degree n

Assume a multivariate homogeneous polynomial of degree n ,namely y ( x ), to be given and find
the inverse multivariate homogeneous polynomial of degree n ,namely x ( y ), and

n

y ( x )=A 11 x +A 12 x [2] + ··· +A 1 n− 1 x [ n− 1] +A 1 n x [ n ] =

A 1 k x [ k ]

p ,

∀ x ∈ R

k =1

(B.60)

n

x ( y )=B 11 y +B 12 y [2] + ··· +B 1 n− 1 y [ n− 1] +B 1 n y [ n ] = B 1 k y [ k ]

p ,

∀ y ∈ R

k =1