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

Table 1

(a)

𝐚𝐢𝐢+𝟏	𝑎𝑖+1𝑖+1=0	𝑎𝑖+1𝑖+1≠0
𝑎𝑖𝑖=0	𝐵11	𝐵12
𝑎𝑖𝑖≠0	𝐵21	𝐵22

(b)

𝐚𝐧𝟏	𝑎11=0	𝑎11≠0
𝑎𝑛𝑛=0	(𝑙,1)(in𝐵11)	(𝑙,𝑙+1)(in𝐵12)
𝑎𝑛𝑛≠0	(𝑛,1)(in𝐵21)	(𝑛,𝑙+1)(in𝐵22)