1 Introduction and main results
A matrix is totally positive if all its minors are positive. These matrices have important applications in diverse areas in mathematics, including approximation theory, combinatorics, cluster algebras, integrable systems, probability, representation theory, and matrix theory
[1, 2, 3, 4, 9, 12, 15, 22]. There is a well known paper by Fomin and Zelevinsky [8] about tests for totally positive matrices. This paper may be regarded as being similar in spirit. Namely, we provide a novel characterization of total positivity via the sign nonreversal property:Theorem A.
A real matrix is totally positive if and only if the following holds:
If , where has nonzero components, , and , then for each , there exists such that
.
This may bring to mind the parallel and widely studied variation diminishing property, which characterize total positivity.
We provide a test for not just one matrix but an entire interval hull of matrices to be totally positive, by reducing it to a finite set of test matrices. Given matrices , recall that their interval hull, denoted by , is defined as follows:
(1) 
If the matrices and are different, then the interval hull contains uncountably many matrices. One of the interesting questions, related to interval hulls of matrices, considered in the literature is the following: Suppose a finite subset of matrices in has some property, say . Does the entire interval hull have the property ? For example, if the matrices and are invertible and (entry wise), then all the matrices in the interval hull are invertible if and only if ( are invertible and) (entry wise) and [19]. An interval hull is said to be of type if all the matrices in are of type . With this background, our second main result can now be stated.
Theorem B.
Let . Then the interval hull is totally positive if and only if the matrix is totally positive for all and .
The notation will be explained at the end of this section.
Remark 1.1.
In this paper we also work out analogues of these results for several other classes of matrices:
Definition 1.1.

A matrix is an matrix if all its principal minors are negative.

An matrix is an almost matrix if all its principal minors of order up to are positive, and the determinant of is negative.

An real matrix is a semipositive matrix
, if there exists a vector
such that . An real matrix is a minimally semipositive matrix, if it is semipositive and no columndeleted submatrix of is semipositive.
matrices have connections to univalence theory (injectivity of differential maps in ) and the Linear Complementarity Problem [17]. Recently in [5], the first author joint work with Tsatsomeros has established an algorithm to detect whether a given matrix is an matrix or not; as well as an algorithm to construct every matrix recursively. The sign nonreversal property for matrices was established in [16, 17]. In this paper, we recap these results, and also prove their analogues for the remaining classes of matrices defined above. We further obtain the interval hull characterization for all of these classes. Our results are summarized as follows:
Matrix Class 
Sign nonreversal  Testing set  
property  for  
Totally positive 
Theorem A  Theorem B  , 
Totally positive 
Theorem 2.3  Theorem 2.4  , 
of order  
Almost matrices of the 

first category with  Theorem 4.3  Theorem 4.4  , 
with respect to  
Almost matrices 
Theorem 4.1  Theorem 4.2  , 
of the second category 

matrices of the first 
[16, Theorem 4.3]  Theorem 3.5  
category with respect to 

matrices of the 
[17, Theorem 2]  Theorem 3.4  
second category 

Semipositive 
N/A  Theorem 4.5  
Minimally semipositive 
N/A  Theorem 4.5  

We conclude by explaining the notation used above.
Definition 1.2.
Fix integers and matrices , with interval hull .

Define the matrices

Given and , define the matrices

If , define .
Organization of the paper.
2 Results for totally positive matrices
In this section, we prove the main theorems above on totally positive matrices. We first develop some preliminary results, and recall past theorems that will be used below.
2.1 Preliminaries
We begin with notation, which will be used throughout the paper without further reference. For a matrix , signifies that all the components of the matrix are nonnegative (positive), and let . Let denote the column vector of size whose entries are all zero. The diagonal matrix with diagonal entries is denoted by . For any positive integer , define . Let . For a subset of , let denote the interior of in with respect to the Euclidean metric. Let denote the vector whose th component is , and other entries are zero.
A square matrix is a matrix if all its principal minors are positive. In [10], the authors established the sign nonreversal property for matrices.
Theorem 2.1 (Sign nonreversal property).
A matrix is a matrix if and only if and for all imply
Using the sign nonreversal property of matrices, in [21], the authors showed that the interval hull of matrices , where , is a matrix, if a finite collection of matrices in are matrices. In [20], the author considered the positive definiteness and the stability of the interval hulls of matrices.
The next result we require is a wellknown theorem of Fekete. Recall that a contiguous submatrix is one whose rows and columns are indexed by sets of consecutive positive integers.
Theorem 2.2 (Fekete’s criterion, [7]).
Let . Then is totally positive if and only if all contiguous submatrices of have positive determinant.
In order to prove our main results above (as well as later results), we also require two basic lemmas. The first is a straightforward verification:
Lemma 2.1.
If , then for all and .
The next lemma extends [21, Theorem 2.1] to interval hulls of arbitrary matrices .
Lemma 2.2.
Let and . Let such that if and if . If , then
Proof.
Let . Then . Since and , so
for all .
Let . For fixed , we have
Hence
Since for each , so . ∎
2.2 Proofs of main results
With the above results in hand, we now show our main theorems.
Proof of Theorem A.
Let be a totally positive matrix, and let , where with for all , , and . Fix , and let be the submatrix of with rows and columns indexed by the sets and , respectively. Then Since is a matrix, there exists such that . Thus
We prove the converse using induction. Let , then . By assumption, for each . Thus , so Thus all the minors are positive. Let be any minor of corresponding to the rows and and the columns and of . Let with . Let , where lies in the positions and . Then for each , there exists such that . Choose , then and . Thus, either or . So is a matrix. Thus all the contiguous minors of are positive.
For the induction step, assume that all contiguous minors of of size less than are positive. Let be a submatrix of corresponding to the rows and the columns . Let with , for and let , where lies in the positions. Then, for , there exists such that . Set . As , so for . Thus for some , and hence is a matrix. Therefore all contiguous submatrices of have positive determinants, and so is totally positive by Theorem 2.2. ∎
Theorem A helps prove our next main result:
Proof of Theorem B.
Let be totally positive for all and , and let . Let , where , and the entries of are nonzero. We claim that, for each , there exists such that Indeed, let be the submatrix of with the rows and the columns indexed by and , respectively. The matrices and are defined similarly. Then, . Since , by Lemma 2.2, there exists such that . Define and as follows:
and
Then is a submatrix of , and hence is a matrix, so for some . That is, , for some . By Theorem A, is totally positive. ∎
Both of our main results have analogues for totally positive matrices of order , for all . Recall that a matrix is called a totally positive matrix of order if all its minors of order at most are positive. The next two theorems give the sign nonreversal property and the characterization of the interval hull for such matrices . The proofs are similar to those of Theorems A and B respectively, and we leave the details to the interested reader.
Theorem 2.3.
An matrix is totally positive of order if and only if the following holds:
If , where has nonzero components, , and , then for each , there exists such that
.
Theorem 2.4.
Let . Then the interval hull is totally positive of order if and only if is totally positive of order for all and .
3 Results for matrices
We now develop similar results to above, for matrices (see Definition 1.1). First recall that an matrix is said to be of the first category if it has at least one positive entry. Otherwise, is of the second category.
The following result gives a characterization for matrices of the second category. This is known as the sign nonreversal property for the matrices of the second category.
Theorem 3.1.
[17, Theorem 2] Let . Then is an Nmatrix of the second category if and only if and does not reverse the sign of any nonunisigned vector, that is, and for all imply or .
For , denotes . For subsets with elements arranged in ascending order, denotes the submatrix of whose rows and columns are indexed by and , respectively.
Theorem 3.2.
[16, Theorem 4.3] Let be an matrix of the first category. Then can be written in the partitioned form (after a principal rearrangement of its rows and columns, if necessary)
(2) 
with , , , and , where is a nonempty proper subset of .
Definition 3.1.
Let be a nonempty proper subset of . An matrix is called an matrix of the first category with respect to if it is of the form with , , , and .
The next result gives a characterization for matrices of the first category with respect to , and is known as the sign nonreversal property for such matrices.
Theorem 3.3.
[16, Theorem 4.3] Let be a nonempty proper subset of and let where , , , and Then is an matrix of the first category with respect to if and only if reverses the sign of a vector , i.e, for all , then either and , or and
We now characterize interval hulls of matrices, beginning with those of the second category.
Theorem 3.4.
Let such that and . Then, is an matrix of the second category if and only if is an matrix of the second category for all .
Proof.
Suppose is an matrix of the second category, then is an matrix of the second category. Conversely, suppose that is an matrix of the second category for all . Let and such that for . By Lemma 2.2, there exists such that , for . By Theorem 3.1, or , since is matrix of the second category. Since , so . By Theorem 3.1, is an matrix of the second category. ∎
We next characterize the interval hull of matrices of the first category with respect to , where .
Theorem 3.5.
Let be a nonempty proper subset of , and let
be two matrices such that , and . Then, is an matrix of the first category with respect to if and only if is an matrix of the first category with respect to for all .
Proof.
Suppose is an matrix of the first category with respect to for all . Since , the block structure of with respect to sign pattern, is independent of and agrees with that of and . Let . Then can be partitioned as where , and . Let such that for all . By Lemma 2.2, there exists such that , for all . Since is an matrix of the first category with respect to , by Theorem 3.3, either and , or and Thus, by Theorem 3.3, is an matrix of the first category with respect to . Conversely, suppose that is an matrix of the first category with respect to . By Lemma 2.1, for all . Thus is an matrix of the first category with respect to for all . ∎
4 Results for almost matrices and semipositive matrices
We now establish the sign nonreversal property for almost matrices (see Definition 1.1) and characterize their interval hull. Recall that an matrix is an almost matrix if and only if is an matrix [13, Lemma 2.4]
. Motivated by this result, we classify almost
matrices into two categories:Definition 4.1.

Let be a nonempty proper subset of . An almost matrix is an almost matrix of the first category with respect to if is an matrix of the first category with respect to .

An almost matrix is an almost matrix of the second category if is an matrix of the second category.
Observe that if is an almost matrix of the second category, then there exists a positive vector such that . Our next result shows a sign nonreversal property for such matrices.
Theorem 4.1.
Let . Then, is an almost matrix of the second category if and only if the following hold:

, where denotes the null space of ,

for all implies that , if for some ; otherwise or ,

.
Proof.
Let be an almost matrix of the second category, and for all . Suppose that for some . Let be the principal submatrix of obtained by deleting the th row and the th column of . Let be the vector obtained from the vector by deleting the th entry. Then for all . Since is a matrix, so for all . Thus Suppose that for all . Let . Then, for all . Since is an matrix of the second category, so, by Theorem 3.1, either or . Note that . Hence, all the components of the vector are either positive or negative. As is an almost matrix of the second category, there exists a positive vector such that . Thus holds.
To prove the converse, first let us show that all the proper principal minors are positive. Let be any principal submatrix of . Without loss of generality, assume that is obtained from by deleting the last row and the last column of . Let , and define . If for all , then for all . Thus, by the assumption, , and hence . By Theorem 2.1, all the proper principal minors of are positive. We now claim is invertible. Indeed, let be a vector such that . Then either or or . Since , so . Also is not a matrix, since , so reverses the sign of a nonzero vector. Thus , hence is an almost matrix. Let . Note that the th entry of the vector is negative. Now, we have . Thus the vector is negative, and hence . So is an almost matrix of the second category. ∎
Using Theorem 4.1, we establish an equivalent condition for an interval hull to be a subset of the set of all almost matrices of the second category.
Theorem 4.2.
Let . Then is an almost matrix of the second category if and only if and are almost matrices of the second category for all .
Proof.
Let be an almost matrix of the second category. Thus, by Lemma 2.1, the matrices and are almost matrices of the second category for all .
Conversely, suppose that and are almost matrices of the second category for all . Let . First let us show that . Let . Then , since and . Since , so , a contradiction. Thus .
Since is an almost matrix of the second category, there exists a vector such that . Thus , since . Hence .
For a nonempty proper subset of , define
(3) 
Note that if is an almost matrix of the first category with respect to , then there exists such that . Next, we develop the sign nonreversal property for almost matrices of the first category with respect to .
Theorem 4.3.
Let and let be a nonempty proper subset of . Then is an almost matrix of the first category with respect to if and only if the following hold:

,

and for all imply if for some ; otherwise either or ,

.
Proof.
Let be an almost matrix of the first category with respect to . Then, , where and . Let If for some , then, by an argument similar to that of Theorem 4.1, we get . So, let us assume that and for each . Let . Then for each . Hence, by Theorem 3.3, where either , or , . Also, , so or As is an almost matrix of the first category with respect to , so there exists such that . Thus holds true.
Conversely, suppose that satisfies all the three properties as above. Let be any principal submatrix of . Without loss of generality, assume is obtained from by deleting the last row and the last column of . Let Set If for all , then for all . Thus, by assumption , so . Thus all the proper principal minors of are positive. Since , so is invertible by the proof of Theorem 4.1. As, , so the matrix is not a matrix. Thus is an almost matrix. Let . Then for all . If , then and , since the th entry of is negative. Otherwise and . Hence is an almost matrix of the first category with respect to . ∎
For a nonempty proper subset of , define the matrices
(4) 
Also define . One can verify that .
In the following theorem, we establish an equivalent condition for an interval hull to be a subset of the set of all almost matrices of the first category with respect to .
Theorem 4.4.
Let . Then is an almost matrix of the first category with respect to if and only if and are almost matrices of the first category with respect to for all .
Proof.
Let be an almost matrix of the first category with respect to . Then the matrices and are almost matrices of the first category with respect to for all .
Conversely, suppose that the matrices and are almost matrices of the first category with respect to for all . From the definition, . Let . First let us show that . Let and . Then and . Now, ,
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