ZPC Matrices and Zero Cycles

Let H be an m × n real matrix and let Zi be the set of column indices of the zero entries of row i of H. Then the conditions |Zk ∩ ∪k−1 i 1 Zi | ≤ 1 for all k 2 ≤ k ≤ m are called the row Zero Position Conditions ZPCs . If H satisfies the ZPC, then H is said to be a row ZPC matrix. If H satisfies the ZPC, thenH is said to be a column ZPCmatrix. The real matrixH is said to have a zero cycle if H has a sequence of at least four zero entries of the form hi1j1 , hi1j2 , hi2j2 , hi2j3 , . . . , hikjk , hikj1 in which the consecutive entries alternatively share the same row or column index but not both , and the last entry has one common index with the first entry. Several connections between the ZPC and the nonexistence of zero cycles are established. In particular, it is proved that a matrix H has no zero cycle if and only if there are permutation matrices P and Q such that PHQ is a row ZPC matrix and a column ZPC matrix.