Sign non-reversal property for totally positive matrices and testing total positivity on their interval hull

We establish a novel characterization of totally positive matrices via a sign non-reversal property. In this we are inspired by the analogous results for P-matrices (matrices with all principal minors positive). Next, the interval hull of two m × n matrices A=(a_ij) and B = (b_ij), denoted by 𝕀(A,B), is the collection of all matrices C such that c_ij=t_ija_ij+(1-t_ij)b_ij for all i,j, where t_ij∈ [0,1]. Using the sign non-reversal property, we identify a finite subset of 𝕀(A,B) that detects the total positivity of all of 𝕀(A,B). This provides a test for an entire class of matrices simultaneously to be totally positive. We also establish analogous results for other classes of matrices: almost P-matrices, N-matrices and (minimally) semipositive matrices.

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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 non-reversal 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.

Both of these results have analogues for totally positive matrices of order (see Theorem 2.3 and 2.4).

In this paper we also work out analogues of these results for several other classes of matrices:

Definition 1.1.
  1. A matrix is an -matrix if all its principal minors are negative.

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

  3. 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 column-deleted 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 non-reversal 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 non-reversal     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  


Table 1: Summary of results. Here, denotes a subset of .

We conclude by explaining the notation used above.

Definition 1.2.

Fix integers and matrices , with interval hull .

  1. Define the matrices

  2. Given and , define the matrices

  3. If , define .

Organization of the paper.

In Section 2, we collect some known results and prove Theorem A and Theorem B. Section 3 contains the results for -matrices and in Section 4, we prove the results for almost -matrices and semipositive matrices.

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 non-reversal property for -matrices.

Theorem 2.1 (Sign non-reversal property).

A matrix is a -matrix if and only if and for all imply

Using the sign non-reversal 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 well-known 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.

For more details about totally positive matrices, we refer to [6, 12, 18].

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 non-reversal 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 non-reversal property for the -matrices of the second category.

Theorem 3.1.

[17, Theorem 2] Let . Then is an N-matrix of the second category if and only if and does not reverse the sign of any non-unisigned 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 non-reversal 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 non-reversal 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 non-reversal property for such matrices.

Theorem 4.1.

Let . Then, is an almost -matrix of the second category if and only if the following hold:

  1. , where denotes the null space of ,

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

  3. .

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 .

Let and such that for . By Lemma 2.2, there exists a vector such that for . Since is an almost -matrix of the second category, by Theorem 4.1, , if for some , otherwise or . Thus, by Theorem 4.1, is an almost -matrix of the second category. ∎

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 non-reversal 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:

  1. ,

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

  3. .

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, ,