#### Abstract

Let be an real matrix and let be the set of column indices of the zero entries of row of . Then the conditions for all are called the (row) Zero Position Conditions (ZPCs). If satisfies the ZPC, then is said to be a (row) ZPC matrix. If satisfies the ZPC, then is said to be a column ZPC matrix. The real matrix is said to have a zero cycle if has a sequence of at least four zero entries of the form 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 has no zero cycle if and only if there are permutation matrices and such that
*PHQ* is a row ZPC matrix and a column ZPC matrix.

#### 1. Introduction

A matrix whose entries are from the set is called a *sign pattern matrix* (or sign pattern). For a real matrix is the sign pattern obtained by replacing each positive (respectively, negative, zero) entry of by (respectively, , 0). For a sign pattern , the * sign pattern class of* is defined by

Further information on sign patterns can be found in [1, 2].

For a real matrix of size , the * bipartite graph* of is the graph with vertex set such that there is an edge between and iff . The * zero bipartite graph* of is the complement of the bipartite graph of .

In [3], the following result is proved.

Theorem 1.1. *Let , and be real matrices such that . Suppose that the zero bipartite graph of (this is the same as the complement of the bipartite graph of ) is a forest. Then there exist rational perturbations , and of , and , respectively, in the same corresponding sign pattern classes, such that *

The purpose of this note is to investigate when the zero bipartite graph of a matrix is a forest from a combinatorial point of view. This will be done in terms of ZPC matrices and zero cycles.

#### 2. ZPC Matrices

*Definition 2.1. *Let be the set of column indices of the zero entries of row () of a real matrix . Then the conditions
are called the *(row) Zero Position Conditions (ZPCs)*. If satisfies the ZPC, then we say that is a (row) *ZPC matrix*. If satisfies the ZPC, then we say that is a *column ZPC matrix*.

*Definition 2.2. *A zero entry in a matrix is called a *covered zero*, if there is another zero entry above this entry in the same column. A zero entry that is not a covered zero is called an * uncovered zero*.

The proposition below follows directly from the definition of the ZPC.

Proposition 2.3. *A matrix satisfies the ZPC if and only if each row of has at most one covered zero.*

*Remark 2.4. *Permutation of columns preserves the ZPC property.

Proposition 2.5. *Let be a ZPC matrix. Then there is a permutation matrix such that each covered zero of is the leading zero in its row.*

*Proof. *We proceed by induction on the number of rows.

The result is trivially true for every matrix with only one row.

Assume that this result holds for ZPC matrices with rows. Now consider any ZPC matrix with rows. Write , where has rows. By induction hypothesis there is a permutation matrix such that each covered zero of is the leading zero in its row.

Since satisfies the ZPC, the last row of has at most one covered zero. If the last row of has no covered zero, then the result is already true based on the induction hypothesis.

Assume that the last row of (or equivalently ) has a covered zero entry. Then all the other zero entries in the last row are uncovered and hence all the entries directly above these zeros in the last row are all nonzero. Therefore, we may permute the columns of to put the columns containing the uncovered zeros of the last row of to the far right positions, resulting in . As none of these columns moved to the far right contains covered zeros of the first rows of , we see that has the property that each covered zero of is the leading zero in its row.

The ZPC places severe restrictions on the location and number of zeros in the matrix. In particular, we have the following result on the number of zeros.

Proposition 2.6. *If is a ZPC matrix, then has at most zeros.*

*Proof. *We proceed by induction on the number of rows.

If , then clearly has at most zeros.

Assume that the result holds for ZPC matrices with rows. Now consider a ZPC matrix with rows.

Suppose that the last row of has zeros. If , then the result follows immediately by applying the induction hypothesis on the submatrix of consisting of the first rows.

Assume that . Since the last row of has at most one covered zero, the last zeros are not covered, and permuting the columns of if necessary (to put the uncovered zeros of the last row to the far right), we may assume that has the block form
where is is and has no zero entry, has one (possibly covered) zero, and consists of uncovered zeros.

The induction hypothesis applied on says that has at most zeros. Hence, has at most zeros.

#### 3. ZPC and Nonexistence of Zero Cycles

In this section we explore interesting connections between the Zero Position Conditions and the nonexistence of zero cycles.

Let be a real matrix. We say that has a * zero cycle* if has a sequence of at least four zero entries of the form
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. For example,
would form a zero cycle of if all these entries are zeros. The idea of a zero cycle (called a loop) was introduced in [4, 5].

The * zero bipartite graph* of a matrix , denoted , has vertex sets and such that there is an edge between and iff . It can be seen that has a zero cycle iff the zero bipartite graph has a cycle.

Theorem 3.1. *If a matrix has a zero cycle, then no (row or column) permutation of is a ZPC matrix or a column ZPC matrix.*

*Proof. *Suppose that has a zero cycle. For any two permutations and of suitable orders, also has a zero cycle. Consider a cycle of . Let be the largest row index involved in the cycle. Then all the zero entries (there are at least two such entries) in the cycle with row index are covered zeros. Hence, is not a ZPC. A similar argument on the columns shows that is not a column ZPC matrix.

We need the following basic fact from graph theory.

Lemma 3.2. *If the degree of every vertex of a graph is at least 2, then has a cycle.*

We are now ready to establish a main result in this section.

Theorem 3.3. *If a matrix has no zero cycle, then there are permutation matrices and such that is both a row ZPC matrix and a column ZPC matrix.*

*Proof. *We proceed by double induction on the number of rows () and the number of columns () of a matrix.

Since every or matrix is a row ZPC matrix and a column ZPC matrix, the result is trivially true for or .

Assume that the result holds for matrices with rows or columns. Consider an matrix . Since the zero bipartite graph of has no cycle, by Lemma 3.2, has a vertex of degree at most 1. Without loss of generality, we may assume that has degree at most 1. This can be achieved by a row permutation on and possibly taking the transpose (in the case when a vertex of degree at most 1 is in ). Since the submatrix of obtained from by deleting the last row clearly has no zero cycle, by induction hypothesis, the rows and columns of may be permuted to produce a matrix that is both a row ZPC matrix and a column ZPC matrix. Of course, the row permutation of does not affect the last row of , while the column permutation of should also be applied to the last column of . For convenience, we may assume that is a row ZPC matrix and a column ZPC matrix.

We now show that with the above mentioned permutations and a possible transposition, the resulting matrix (also denoted by for convenience) is a row ZPC matrix and a column ZPC matrix. Since the last row of contains at most one zero entry, and hence at most one covered zero, while is a row ZPC matrix, it follows immediately that is a row ZPC matrix.

Observe that the last column of (the transpose of the last row of ) contains at most one zero entry, and hence, there is no covered zero in the last column of . Combined with the assumption that is a row ZPC matrix, we see that each row of has at most one covered zero and so is a row ZPC matrix, namely, is a column ZPC matrix.

By Theorem 3.1, the converse of Theorem 3.3 is true. Thus, we also have the following theorem.

Theorem 3.4. *A matrix has no zero cycle iff there are permutation matrices and such that is a row ZPC matrix and a column ZPC matrix.*

We now come to the culminating result.

Theorem 3.5. *The following statements are equivalent. *(i)* has no zero cycle.*(ii)*The zero bipartite graph of has no cycle.*(iii)*The zero bipartite graph of is a forest.*(iv)*There is a permutation matrix such that is a row ZPC matrix.*(v)*There is a permutation matrix such that is a column ZPC matrix.*(vi)*There are permutation matrices and such that is both a row ZPC matrix and a column ZPC matrix.*

*Proof. *It can be seen that the first three statements are equivalent. From Theorem 3.4, statements (i) and (vi) are equivalent. Since permutation of columns (rows) preserves the row ZPC (column ZPC) property, it is clear that statement (vi) implies each of the statements (iv) and (v). Next, suppose that has a zero cycle. Then has a zero cycle for any permutation matrix . Hence, the last row of that has at least two zero entries of the above zero cycle of contains at least two covered zeros, so that is not row ZPC. Thus, statement (iv) implies statement (i). Similarly, statement (v) implies statement (i). The proof is now complete.

We point out that Proposition 2.5 also follows from Theorem 3.5, since the number of edges of a forest with vertices is at most .

The results of this paper may be stated in terms of the positions of the nonzero entries of a matrix and the bipartite graph of the matrix. However, we chose to use the zero bipartite graph because the motivation [3] for this study naturally requires concentrating on the zero entries.