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Page 8 of 18 Guo et al. Intell Robot 2023;3(4):596-613 I http://dx.doi.org/10.20517/ir.2023.32

On the other hand, since matrix is related to the strongly connected graph, and according to Lemma 2,
is an irreducible matrix. Therefore, according to the Perron-Frobenius theorem, vector is a positive vector.
Then, the vector to be found is expressed as

Õ
(19)
= / | |
=1

The proof is, thus, completed.

According to Lemma 4, the positive vector = [ 1 , 2 , . . . , ] is defined as corresponding to matrix −1 in
system (16), where Í = 1. Define Γ = ( 1 , 2 , 3 , . . . , ), and Γ is an invertible matrix. Then, one
=1
has inferences as follows:

[1, 1, . . . , 1]Γ −1 = 0 (20)

Γ −1 [1, 1, . . . , 1] = 0 (21)

Furthermore, system (13) is equivalent to the system as follows:

−1
Γ ⊗ × + Γ −1 ⊗ × + Γ ⊗ × = 0 (22)
¤
¥
Therefore:

1 0

Γ = (23)
0 Γ
e
Similarly, one has that

0 0
−1

Γ = (24)
0
e
" #

0 e
−1
Γ = (25)
0
e
Define = ( 1 , ), where = ( 2 , 3 , . . . , ). System (22) is equivalent to the system as follows:

" #

¤
1 0 1 0 0 1 0 e 1
¥
⊗ × + ⊗ × + ⊗ × = 0 (26)
¥
0 Γ 0 0
¤
e
e
e
From system (26), together with (5), one has that
¤
( ) = ( )
(27)
¤ ( ) = [ 1 2 ] ⊗ ( ) ∈ [ , +1 )