Characterization of the Existence of an N 0-Completion of a Partial N 0-Matrix with an Associated Directed Cycle

An n × n matrix is called an N 0-matrix if all its specified principal minors are nonpositive. In the context of partial matrices, a partial matrix is called a partial N 0-matrix if all its specified principal minors are nonpositive. In this paper we characterize the existence of an N 0-matrix completion of a partial N 0-matrix whose associated graph is a directed cycle.


Introduction
A partial matrix is a rectangular array, some of whose entries are specified while the remaining unspecified entries are free to be chosen (from a certain set). In this paper we are going to work on the set of the real numbers and to assume that all diagonal entries are prescribed. A completion of a partial matrix is the matrix resulting from a particular choice of values for the unspecified entries. A completion problem asks if we can obtain a completion of a partial matrix with some prescribed properties.
The technics to obtain this completion depend on the pattern of the partial matrix which can be combinatorially symmetric (i.e., is specified if and only if is) or noncombinatorially symmetric. Here we are going to work with this second class of partial matrices.
A natural way to describe an × partial matrix = ( ) is via a graph = ( , ), where the set of vertices is {1, 2, . . . , } and there is an arc from to if and only if position ( , ) of is specified. In general, a directed graph (resp., nondirected graph) is associated with a noncombinatorially symmetric (resp., combinatorially symmetric) partial matrix. Since all main diagonal entries are specified we omit loops.
As a class of square real matrices that contains thematrices we define the 0 -matrices, × real matrices = ( ), where all its principal minors are nonpositive. Since 0matrices are preserved by principal submatrices we define a partial 0 -matrix as a partial matrix whose completely specified principal submatrices are 0 -matrices.
In general it is not always true that a partial 0 -matrix has an 0 -matrix completion as the following matrix shows (see [7]): is a partial 0 -matrix that has an 0 -matrix completion that leads us to analyze the 0 -matrix completion problem depending on the pattern of the partial matrix. We have studied in [7] when a combinatorially symmetric partial 0matrix with no null main diagonal entries such that the graph of its specified entries is a 1-chordal graph or a cycle has an 2 The Scientific World Journal In this paper we study the mentioned problem for partial 0 -matrices that can have zeros at the main diagonal and whose associated graph is a directed cycle of length equal to the order of the matrix. In this case we may suppose without loss of generality that these matrices have the form: where denotes a specified entry and an unspecified one. Observe that when we say " , +1 , = 1, 2, . . . , , where the subscripts are expressed module ", we are using the congruence module ; that is, we are considering the entries 12 , 23 , . . . , −1 , 1 .
Given a matrix of size × the submatrix lying in rows and columns , , ⊆ = {1, 2, . . . , } is denoted by We denote = 1 if = 0 and = if ̸ = 0. In the next section we introduce necessary and sufficient conditions in order to guarantee the existence of an 0matrix completion of a partial 0 -matrix whose associated graph is a directed cycle.

Completion of Partial 0 -Matrices
It is easy to prove that 0 -matrices as well as partial 0matrices satisfy the following properties.
(1) If is a permutation matrix, then is an 0matrix.   In [7] the authors proved that any × 0 -matrix with no null diagonal entries is diagonally similar to an 0 -matrix in the set: But since there are 0 -matrices with some entries equal to zero we need to introduce the following set: We also extend the definition of matrices to partial matrices; that is, consists of the × partial matrices = ( ) such that if ̸ = 0 then sign( ) = (−1) + +1 , for all specified entry ( , ), , ∈ {1, 2, . . . , }. The following matrix is an example of a matrix of 4 , ] .
The following results, consequence of Proposition 1, allow us to transform a partial 0 -matrix = ( ), whose associated graph is a directed cycle, into a matrix whose diagonal nonzero values are −1; the nonzero elements of the first upperdiagonal are 1 and the element in position ( , 1) is There exists a positive diagonal matrix such that matrix = = ( ) is also an 0 -partial matrix with equal to −1 or to zero, for all = 1, 2, . . . , .
Proof. It suffices to consider = diag(1, 12 , 12 23 , . . . , Therefore, if = ( ) is an × partial 0 -matrix whose associated graph is a directed cycle, we will assume, without loss of generality, that has the following structure: −1 or zeros on the main diagonal, 1's or zeros in the first upper diagonal and 12 23 ⋅ ⋅ ⋅ 1 in position ( , 1).
The following theorem characterizes the matrices as an intermediate step to obtain the desired completion. It can be easily obtained from the transformations of Propositions 2 and 3.
Now we analyze the existence of an 0 -matrix completion of a partial 0 -matrix with an associated directed cycle, by distinguishing between matrices with no null main diagonal entries and matrices with some null values in the main diagonal.

Theorem 5.
Let be an × partial 0 -matrix, with nonzero main diagonal entriessuch that its associated graph is a directed The Scientific World Journal 3 cycle. The following statements are equivalent: (2) is diagonally similar to an element of , Proof. Observe that from (4) of Proposition 1, we have that all the specified entries are nonzero. Then, from commentary after Proposition 3, we assume that all the elements in the main diagonal are −1 and the first upper diagonal is formed by 1's. Let us suppose that is even; the case odd is analogous. Since the upper diagonal and the element in position ( , 1) are nonzero, by applying Theorem 4, the condition Now, we assume that the second statement is true. We consider = ( ), where = if is a specified value of , +1 = 1 for = 1, 2, . . . , , where subscripts are expressed module and 1 = 1/( 12 23 ⋅ ⋅ ⋅ 1 ). Then is an × partial 0 -matrix, with nonzero main diagonal entries such that its associated graph is a nondirected cycle. Theorem 4.3 of [7] assures that , and therefore has an 0 -matrix completion.
Finally, from the note after Proposition 1, the third statement implies the second one. Now, it arises the question about establishing an analogous result to Theorem 5, when zero entries appear in the main diagonal. The answer is negative since if we admit a zero diagonal element and a zero entry in the upper diagonal, there exist matrices in , is even, such that 12 23 ⋅ ⋅ ⋅ 1 is negative, but that admits an 0 -matrix completion.
The following results characterize this type of matrices. Note that, if there are some null main diagonal entries, the existence of an 0 -matrix completion implies that if +1 +1 ̸ = 0, then +1 ̸ = 0. So, we add this condition as a hypothesis. In addition, recall that we are going to assume that = −1 or zero for all ∈ {1, 2, . . . , }; the entries in the first upper diagonal are 1 or zero and the value of the element in position ( , 1) is 12 23 ⋅ ⋅ ⋅ 1 .
Note that elements of the first upper diagonal +1 , ∈ {1, 2, . . . , − 1}, are moved to blocks , , ∈ {1, 2}, depending on the value of and +1 +1 : if both entries are zero, after the permutation +1 will be in 11 ; if = 0 and +1 = 1 the element +1 will be in 12 ; if = −1 and +1 = 0 then +1 will be in 21 and in other cases +1 will be in 22 . This is shown in Table 1(a). In Table 1(b) we can see the position that the element 1 of will occupy after the permutation.
Then we can be sure that each of the first lines of the permutated matrix will have as maximum a nonzero value and the last ones will have exactly one −1 and only another nonzero value as maximum.
We complete with zeros the unspecified entries of blocks 11 , 12 , and 21 . In order to complete 22 we distinguish two cases.
The completion of , , is an 0 -matrix since (a) the principal minors lying rows and columns with indices in are zero, as we can prove by developing by the nonzero elements (there is as maximum one nonzero entry by line); (b) the principal minors lying rows and columns with indices in are less than or equal to zero, since 4 The Scientific World Journal (c) the principal minors with rows and columns with indices in and are zero, because of they have a row of zeros or, as before, by developing by the only nonzero element of each row, the minor is reduced to one of submatrix 21 .
So, matrix has an 0 -matrix completion.
Proof. Let be even. If 12 23 . . . 1 > 0 some changes in the proof of the above theorem gives the desired completion. Specifically, choose the zero diagonal entry with less index. If 1 ≤ ≤ − 1 since 12 23 ⋅ ⋅ ⋅ 1 > 0 matrix can be transformed by diagonal similarity in a matrix such that if is even, 1 = 1 and +1 are equal to a positive value and if is odd, 1 = −1 and +1 are equal to a negative value. By permutation similarity we get a block-matrix analogous to the previous result and by a similar reasoning we get that there exists an 0 -matrix completion of . If = we transform by the permutation = [ , 1 , . . . , −2 , −1 ], where is the canonical vector for all ∈ {1, 2, . . . , } and we proceed in a similar way to the previous result. Now, we are going to prove the necessity of the condition by induction on ; that is, if there exists an 0 -completion of , = ( ), then 12 23 ⋅ ⋅ ⋅ 1 > 0 if is even and 12 23 ⋅ ⋅ ⋅ 1 < 0 if is odd. We denote = 12 23 ⋅ ⋅ ⋅ 1 . For = 3, if we suppose that > 0 by analyzing the seven different cases that arise depending on the number of zeros, one, two, or three in the main diagonal, we get that det > 0, that is a contradiction. Then < 0.
Let > 3 be. Suppose is even (if is odd the process is analogous). We are showing that in the mentioned conditions if there exists an 0 -completion of , then > 0. Let be an 0 -completion of .
Let us suppose by hypothesis of induction (HI) that the statement is true for all 3 ≤ ≤ − 1. Since is an 0matrix, we obtain, from the 2×2 principal minors, that all the entries of the first under diagonal are greater than or equal to zero; that is, −1 ≥ 0 for all ∈ {1, 2, . . . , }.
In can be considered as a completion of a partial 0 matrix of size strictly smaller than with all the first upper diagonal formed by 1's. If has at least a zero diagonal entry, taking into account that 1 ≤ 0 if is odd and 1 ≥ 0 if is even, the hypothesis of induction allows us to assure that the entry in position ( − + 2, 1) of ; that is, is positive. In the other case, if all the diagonal entries are nonzero we get the same conclusion by Theorem 5. This ends the proof in this case. In this case 1 = 0 for all ∈ {3, . . . , } if some entry of the upper triangular part of is nonzero, we also obtain that > 0 by using det [{1, 2, . . . , −1, , , +1, . . . , }] ≤ 0 and HI.
Therefore, it remains to analyze the case in which all the upper triangular part except the first upper diagonal is zero. As we will see now, most of the cases can not be given.
Let be the nonzero diagonal entry with less index. If  ∃ 0 -matrix completion (Theorem 6) is an × , n even, partial 0 -matrix, whose associated graph is a directed cycle. We sum up the results of Theorems 6 and 7 in Table 2. One can consider a similar one for odd.