Characterization of the Existence of an 𝑁₀-Completion of a Partial 𝑁₀-Matrix with an Associated Directed Cycle

Table 2

*Under the assumption that 𝑎_𝑖𝑖𝑎_𝑖+1𝑖+1≠0 implies 𝑎_𝑖𝑖+1≠0.

	∀𝑖𝑎𝑖𝑖+1≠0	∃𝑖𝑎𝑖𝑖+1=0^∗
∀𝑖𝑎𝑖𝑖≠0	𝑎12𝑎23…𝑎𝑛1>0 iff; ∃𝑁0-matrix completion; (Theorem 5)	𝑁0-matrix completion has no sense; (Proposition 1)
∃𝑖𝑎𝑖𝑖=0	𝑎12𝑎23…𝑎𝑛1>0 iff; ∃𝑁0-matrix completion; (Theorem 7)	∃𝑁0-matrix completion; (Theorem 6)

𝐴 is an 𝑛×𝑛, 𝐧 even, partial 𝑁₀-matrix, whose associated graph is a directed cycle.