To obtain the explicit expression of Cramér-Rao lower bound (CRLB) and simplify the inversion process of higher-order Fisher information matrices (FIM) for undirected network navigation of multi-agent systems (MAS) with arbitrary information coupling

we proposed a distributed inversion method named block diagonalization method. Firstly

a high-order FIM was constructed for the arbitrary coupling undirected network navigation and positioning model

and then on the basis of its symmetry characteristics and the relationship between submatrices and network topology

the FIM was expressed as a linear combination of two block diagonal matrices

which contained the Fisher information of all nodes and edges respectively. Secondly

the matrix inversion lemma was used twice to derive the explicit expression of the inverse of FIM

i.e. CRLB and the equivalent FIM for network nodes. An iterative process was also introduced to decompose the whole inversion process into computations between several lower-order matrices

so as to mitigate the computational burden at every step. Finally

the proposed algorithm was verified by numerical experiments

and compared with the matrix block iterative inversion method. Results show that both methods had high computational efficiency and accuracy

but the computational burden of the proposed method was lower and the computational speed was faster

which verifies its accuracy and effectiveness. This algorithm can be used to analyze the Fisher information fusion process for every node or edge in arbitrary coupling undirected network navigation and positioning model.