Positive Hankel Matrices, Eigenvalues and Total Positivity

Abstract

For positive Hankel matrices, an interval containing all eigenvalues is obtained. With a stronger condition, we also construct two sharper intervals for the eigenvalue localization. The total positivity of positive Hankel matrices is analyzed. The relationship between Hankel TP matrices and some Toeplitz SR matrices is analyzed.

1. Introduction

Hankel matrices are matrices constant along their antidiagonals (see Section 2). They were named after the German mathematician Hermann Hankel. They were originally studied due to their elegant algebraic properties (cf. [1]) and their relationship with Hankel operators (cf. [2]), and later due to their important applications to many fields, such as signal processing, control theory, quantum mechanics and cryptography. For instance, in signal processing, the matrix structure of Hankel matrices makes them useful to represent the autocorrelation and cross-correlation of signals, as well as to analyze the spectral properties of signals. In control theory, the matrix structure of Hankel matrices makes them useful to represent the the input–output behavior of a system and to analyze and design control systems. Another class of matrices with important applications in many fields is the class of totally positive matrices, which lead to a framework of total positivity. A totally positive matrix is a matrix whose minors are all nonnegative (see Section 2 for applications of these matrices). The relationship of Hankel matrices with the class of totally positive matrices (matrices with all their minors nonnegative) is a research field with interesting results. A related concept is given by coefficientwise totally positive matrices, where coefficientwise Hankel totally positive matrices play a key role (see [3] and references therein). Another connection comes from the fact that the Hadamard product of Hankel strictly totally positive matrices is also strictly totally positive (cf. p. 123 of [4] and [5]). In addition, we note that Hankel totally positive matrices have been found in many fields, in particular (cf. [6]), in the problem of finding algorithms with high relative accuracy (cf. [7,8,9]). Algorithms with high relative accuracy have been found for some subclasses of TP matrices [6,10,11,12] in order to compute the inverse, all singular values, all eigenvalues or even the solution of some linear systems with high relative accuracy. This was obtained in [6] for Hankel matrices that were Gramian matrices with respect to inner products defined for Hilbert spaces supported on bounded and unbounded intervals. Particular cases are Hilbert matrices that are well known in numerical linear Algebra.

A first goal of this paper is the presentation of some new contributions on the relationship of Hankel matrices and totally positive matrices. A second goal is related to the problem of the localization of eigenvalues of positive Hankel matrices. The localization of eigenvalues of a matrix is a classical subject, which is still actively considered (cf. [13,14,15,16]). Eigenvalue localization results are very useful for providing information on the eigenvalues as well providing an initial step for many numerical methods. These studies were performed in [17] for Toeplitz matrices. Here, we perform the studies and observe that we obtain different results, which is not surprising because it is known that Hankel matrices have different properties from those of Toeplitz matrices.

The layout of the paper is as follows. In Section 2, we introduce the classes of matrices considered in this paper as well as some basic notations. In Section 3, we first construct an interval containing all the eigenvalues of positive Hankel matrices. With an extra condition, we also construct two intervals with an empty intersection inside the previous one and that also include all the eigenvalues. These two intervals can be considerably small and so very useful for eigenvalue localization. As we illustrate with a family of matrices, both intervals are required and they can be tight for the localization of the eigenvalues.

Section 4 analyzes the connections between positive Hankel matrices and totally positive matrices. We show that the strict total positivity of an $(n+1)\times (n+1)$ Hankel matrix can be characterized by the positivity of $2n+1$ minors. We also obtain a necessary condition for the total positivity of order 2 based on the parameters of a positive Hankel matrix. A very simple condition on the parameters that implies strict total positivity is also presented. We also relate positive Hankel matrices with a class of sign regular Toeplitz matrices. Finally, in Section 5, we summarize the results and conclusions of this paper.

2. Basic Definitions and Notations

In this section, we define and give notations for the classes of matrices that will be considered in this work. We say that a matrix $A={\left(a_{ij}\right)}_{1\le i,j\le n}$ is positive if $a_{ij}>0$, and nonnegative if $a_{ij}\ge 0$ for all $i,j$.

Given a sequence of $2n+1$ real numbers ${\left\{b_k\right\}}_{k=0}^{2n}$, the $(n+1)\times (n+1)$ matrix $A={\left(b_{i+j}\right)}_{0\le i,j\le n}$ is called a Hankel matrix. That is,

$$A=\left(\begin{array}{ccccc}{b}_0& b_1& \cdots & b_{n-1}& b_n\\ b_1& b_2& \cdots & b_n& b_{n+1}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ b_{n-1}& b_n& \cdots & b_{2n-2}& b_{2n-1}\\ b_n& b_{n+1}& \cdots & b_{2n-1}& b_{2n}\end{array}\right).$$

The numbers ${\left\{b_k\right\}}_{k=0}^{2n}$ are called the parameters associated to the Hankel matrix A. Observe that Hankel matrices are symmetric matrices and so their eigenvalues are all real.

We say that a matrix with real entries is totally positive (TP) when all its minors are nonnegative (see [4,18,19,20]). TP matrices are also called totally nonnegative matrices and they have important applications in many fields, such as approximation theory, finance, differential equations, combinatorics, economics, statistics, mechanics, quantum groups, Markov chains, biomathematics and computer-aided geometric design (cf. [4,18,19,21,22,23,24,25,26,27,28,29]). A nonnegative matrix is called TP₂ (or TP of order 2) if all its $2\times 2$ submatrices have a nonnegative determinant. In [19], several properties and applications of TP₂ matrices are presented. If a matrix has all its minors positive, then it is called strictly totally positive (STP).

TP matrices also belong to the more general class of sign regular matrices, which we now introduce. A vector of signs $\epsilon =({\epsilon }_1,\dots ,{\epsilon }_n)$ with ${\epsilon }_j\in \{\pm 1\}$ for $j\le n$ is called a signature. An $n\times n$ matrix A is sign regular (SR) with signature $\epsilon$ if, for each $k=1,\dots ,n$, all minors of order k have the same sign ${\epsilon }_k$ or are zero. If A is SR with signature $\epsilon$ and all its minors are nonzero, then A is called strictly SR (SSR). Sign regular matrices arise in many applications because, in the nonsingular case, they characterize the variation diminishing transformations (cf. [18]).

The last class of matrices that will be mentioned in this paper is the class of Toeplitz matrices, which satisfy that all their diagonals are constant. Given a sequence of $2n-1$ real numbers ${\left\{a_k\right\}}_{k=-n+1}^{n-1}$, an $n\times n$ Toeplitz matrix $A={\left(a_{ij}\right)}_{1\le i,j\le n}$ is defined by $a_{ij}:=a_{i-j}$, for all $1\le i,j\le n$. Toeplitz matrices have been applied to several fields, such as differential and integral equations, signal processing, statistics and physics (cf. [30,31])

We now introduce some matrix notations. For positive integers $k,n$, $1\le k\le n$, $Q_{k,n}$ will denote the set of all increasing sequences of k natural numbers less than or equal to n. We also define the dispersion $d\left(\alpha \right)$ of $\alpha \in Q_{k,n}$ by $d\left(\alpha \right):={\sum }_{i=1}^{k-1}\left({\alpha }_{i+1}-{\alpha }_i-1\right)={\alpha }_k-{\alpha }_1-(k-1),$ with the assumption $d\left(\alpha \right)=0$ for $\alpha \in Q_{1,n}$. Let A be a real $n\times m$ matrix. For $k\le n$, $l\le m$, and for any $\alpha \in Q_{k,n}$ and $\beta \in Q_{l,m}$, $A\left[\alpha \right|\beta ]$ denotes the $k\times l$ submatrix of A containing rows numbered by $\alpha$ and columns numbered by $\beta$. A principal submatrix will be denoted $A\left[\alpha \right]:=A\left[\alpha \right|\alpha ]$ and the leading principal submatrix of order k is given by the submatrix $A[1,2,\dots ,k]$. An initial submatrix is either of the form $A\left[\alpha \right|1,2,\dots ,k]$ with $d\left(\alpha \right)=0$ or of the form $A[1,2,\dots ,k|\beta ]$ with $d\left(\beta \right)=0$. Using determinants, these definitions of submatrices lead to the corresponding definitions with minors.

3. Eigenvalue Localization of Positive Hankel Matrices

We start this section by introducing some basic notations useful for eigenvalue localization.

If $A={\left(a_{ij}\right)}_{1\le i,j\le n}$ is a square matrix, we recall the known Gerschgorin circles (see [16])). We define the ith row Gerschgorin circle as

$$R_i:=\{z\in \mathbf{C}||z-a_{ii}|\le \sum _{k=1}^n|a_{ik}\left|\right\}$$

for each $i=1,\dots ,n$. The column Gerschgorin circles can be defined in a similar way, but using columns instead of rows. By Gerschgorin’s theorem, all eigenvalues of a matrix lie in the union of all row Gerschgorin circles as well as in the union of all column Gerschgorin circles. As we have noted in a previous section, Hankel matrices are symmetric, and so their eigenvalues are real. Therefore, and taking into account that we consider symmetric and positive matrices, the row Gerschgorin circles reduce to the following intervals containing all eigenvalues of A: for each $i=1,\dots ,n$, the interval

$$[a_{ii}-\sum _{k\ne i}{a}_{ik},a_{ii}+\sum _{k\ne i}{a}_{ik}].$$

(1)

In [32], alternative inclusion regions were found for the real parts of the eigenvalues, and they can improve the information provided by Gerschgorin circles in a complementary way. Let us denote for each $i=1,\dots ,n$

$$r_i^{+}:=max\{0,a_{ij}\mid j\ne i\},r_i^{-}:=min\{0,a_{ij}\mid j\ne i\}$$

(2)

and

$$r_i:=\left\{\begin{array}{cc}{r}_i^{+},\hfill & \mathrm{if}{a}_{ii}>0,\hfill \\ r_i^{-},\hfill & \mathrm{if}{a}_{ii}<0.\hfill \end{array}\right.$$

(3)

The following result corresponds to Proposition 1 of [17].

Proposition 1.

Given a symmetric positive matrix $A={\left(a_{ik}\right)}_{1\le i,k\le n}$ and $r_i^{+},r_i^{-},r_i,$ defined in (2), (3), all eigenvalues of A belong to the intersection between the union of row Gerschgorin intervals and the union of the intervals

$$[a_{ii}-r_i^{+}-\sum _{k\ne i}|r_i^{+}-a_{ik}|,a_{ii}-r_i^{-}+\sum _{k\ne i}|r_i^{-}-a_{ik}|],i=1,\dots ,n.$$

(4)

Moreover, for each $i=1,\dots ,n$, $r_i=r_i^{+}>0$, $r_i^{-}=0$.

In contrast to the previous inclusion intervals for the eigenvalues, we now consider an exclusion interval for the real eigenvalues of a real matrix presented in [33]. From now on, for a real matrix $A={\left(a_{ik}\right)}_{1\le i,k\le n}$ and for each $i=1,\dots ,n$, we denote

$$s_i^{+}:=max\{0,min\{a_{ij}\mid j\ne i\}\},s_i^{-}:=min\{0,max\{a_{ij}\mid j\ne i\}\}.$$

(5)

Proposition 2.6 of [33] provides the mentioned exclusion interval.

Theorem 1.

Given a real matrix $A={\left(a_{ik}\right)}_{1\le i,k\le n}$ and $s_i^{+},s_i^{-}$ of (5), we have for any real eigenvalue λ of A that

$$\lambda \notin \left(\underset{i=1,\dots ,n}{max}\left\{\sum _{j=1}^n{a}_{ij}-n{s}_i^{+}\right\},\underset{i=1,\dots ,n}{min}\left\{\sum _{j=1}^n{a}_{ij}-n{s}_i^{-}\right\}\right).$$

(6)

The previous interval of (6) will be named a row exclusion interval.

Replacing A by $A^T$, we can derive an exclusion column interval and derive an analogous result to the previous Theorem 1 for an exclusion column interval.

The following result corresponds to Proposition 2 of [17].

Proposition 2.

Given a symmetric positive matrix $A={\left(a_{ik}\right)}_{1\le i,k\le n}$ and $s_i^{+},s_i^{-}$ of (5), we have that, for each $i=1,\dots ,n$, $s_i^{+}>0$ and $s_i^{-}=0$, and the eigenvalues of A cannot belong to the interval

$$\left(\underset{i=1,\dots ,n}{max}\left\{\sum _{j=1}^n{a}_{ij}-n{s}_i^{+}\right\},\underset{i=1,\dots ,n}{min}\left\{\sum _{j=1}^n{a}_{ij}\right\}\right).$$

(7)

Let us show first a very simple example showing the usefulness of Proposition 2. Let us consider the positive Hankel matrix with parameters equal to a positive real number r. So, A is the matrix with k at any entry:

$$A=\left(\begin{array}{cccc}r& r& \cdots & r\\ r& & & ⋮\\ ⋮& & & r\\ r& \cdots & r& r\end{array}\right).$$

The eigenvalues of A are $nr$ and 0. Then, $s_i^{+}=r$ for all I and the exclusion interval (7) is $(0,nr)$.

For general positive Hankel matrices, we shall combine the previous exclusion interval with some inclusion intervals in order to derive localization regions for the eigenvalues that can be very sharp.

Let us now take a positive Hankel matrix A with parameters ${\left\{b_k\right\}}_{k=0}^{2n}$. We need the next notations:

$$\tilde{s}:=\underset{0\le i\le n}{min}\left\{b_{2i}\right\},s:=\underset{1\le i\le 2n-1}{min}\left\{b_i\right\},S:=\underset{1\le i\le 2n-1}{max}\left\{b_i\right\},$$

(8)

and also these notations indicating the minimal and maximal row sums of A:

$$\sigma :=\underset{0\le i\le n}{min}\{\sum _{k=i}^{i+n}{b}_k\},\Sigma :=\underset{0\le i\le n}{max}\{\sum _{k=i}^{i+n}{b}_k\}.$$

(9)

The next result has two parts. It gives an eigenvalue inclusion interval for all eigenvalues of a positive Hankel matrix and, in the second part, with an extra restriction, it also presents two disjoint intervals included in the previous one and such that all the eigenvalues belong to them. As we later show, both intervals are required and they can give tight localization for the eigenvalues of A.

Theorem 2.

If $A={\left(a_{ij}\right)}_{1\le i,j\le n+1}$ is a positive Hankel matrix with parameters ${\left\{b_k\right\}}_{k=0}^{2n}$ and $\tilde{s},s,S,\sigma ,\Sigma$ are defined by (8) and (9), then all eigenvalues of A belong to the interval

$$[\max \{\sigma -(n+1)S,2\tilde{s}-\Sigma \},\Sigma ].$$

(10)

Furthermore, if we also have $\Sigma -(n+1)s<\sigma$, then all eigenvalues of A are contained in the following union of two disjoint intervals

$$[\max \{\sigma -(n+1)S,2\tilde{s}-\Sigma \},\Sigma -(n+1)s]\cup [\sigma ,\Sigma ].$$

(11)

Proof. 

Taking into account Proposition 1, the eigenvalues of A belong to the intersection between the union of row Gerschgorin intervals and the union of the intervals given by (4). Observe that the right endpoint of each interval of (4) is the same right endpoint of the corresponding row Gerschgorin interval. Due to the positivity of A, we have that the right endpoints of all intervals of (4) coincide with the corresponding row sums and so they are less than or equal to Σ. Taking into account again the positivity of A, each ith left endpoint of each row Gerschgorin interval is provided by $2b_{2(i-1)}$ minus the sum of the entries of the corresponding row and so they are greater than or equal to $2\tilde{s}-\Sigma$. Once again, by the positivity of A, the left endpoints of all intervals of (4) are provided by the corresponding row sum minus $(n+1)r_i^{+}$ and, since the row sum is greater than or equal to $\sigma$ and $r_i^{+}\le S$ for all $i=1,\dots ,n$, we deduce that all left endpoints of all intervals of (4) are greater than or equal to $\sigma -(n+1)S$.

Let us now consider the second part of the statement. Hence, we also suppose that we have $\Sigma -(n+1)s<\sigma$. Given an eigenvalue $\lambda$ of A,we have by Proposition 2 that

$$\lambda \notin \left(\underset{i=1,\dots ,n+1}{max}\left\{\sum _{j=1}^{n+1}{a}_{ij}-(n+1)s_i^{+}\right\},\sigma \right).$$

(12)

Due to the fact that $s\le s_i^{+}$ for all $i=1,\dots ,n+1$, we have that ${\sum }_{j=1}^{n+1}{a}_{ij}-(n+1)s_i^{+}\le {\sum }_{j=1}^{n+1}{a}_{ij}-(n+1)s$ for all $i=1,\dots ,n+1$. Thus,

$$\underset{i=1,\dots ,n+1}{max}\left\{\sum _{j=1}^{n+1}{a}_{ij}-(n+1)s_i^{+}\right\}\le \underset{i=1,\dots ,n+1}{max}\left\{\sum _{j=1}^{n+1}{a}_{ij}-(n+1)s\right\}$$

and, since

$$\underset{i=1,\dots ,n+1}{max}\left\{\sum _{j=1}^{n+1}{a}_{ij}-(n+1)s\right\}=\underset{i=1,\dots ,n+1}{max}\left\{\sum _{j=1}^{n+1}{a}_{ij}\right\}-(n+1)s=\Sigma -(n+1)s,$$

We conclude that condition (12) is transformed into

$$\lambda \notin \left(\Sigma -(n+1)s,\sigma \right).$$

(13)

Taking into account our hypothesis, the exclusion interval of (13) is nonempty and the result is proved. □

The next example provides a family of matrices depending on $\epsilon >0$ such that the two intervals of (11) converge to the two eigenvalues of the limit matrix of this family as $\epsilon \to 0$. This shows that both intervals are necessary and also the potential sharpness of our result.

Example 1.

Let us take the family of Hankel matrices $A_{\epsilon }$ with $2n+1$ positive parameters $\{1-n\epsilon ,1-(n-1)\epsilon ,\dots ,1-\epsilon ,1,1-n\epsilon ,\dots ,1-\epsilon ,\}$ and $0<\epsilon <{\displaystyle \frac{1}{n}}$, that is,

$$A_{\epsilon }=\left(\begin{array}{ccccc}1-n\epsilon & 1-(n-1)\epsilon & \cdots & 1-\epsilon & 1\\ 1-(n-1)\epsilon & 1-(n-2)\epsilon & \cdots & 1& 1-n\epsilon \\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 1-\epsilon & 1& \cdots & 1-3\epsilon & 1-2\epsilon \\ 1& 1-n\epsilon & \cdots & 1-2\epsilon & 1-\epsilon \end{array}\right).$$

As $\epsilon \to 0$, matrices $A_{\epsilon }$ converge to the matrix with all entries equal to 1, whose eigernvalues are 0 and n. For matrices $A_{\epsilon }$, it can be checked that the values of σ and Σ given by (9) are $\sigma =\Sigma =n+1-{\displaystyle \frac{n(n+1)\epsilon }{2}}$. This number is an eigenvalue for all matrices $A_{\epsilon }$ associated to the eigenvector ${(1,\dots ,1)}^T$. Therefore, the second interval $[\sigma ,\Sigma ]$ of Theorem 2 coincides with this eigenvalue. Moreover, for all matrices $A_{\epsilon }$, we have that $\max \{\sigma -(n+1)S,2\tilde{s}-\Sigma \}=\sigma -(n+1)S$. Hence, the first interval of Theorem 2 is

$$[\sigma -(n+1)S,\Sigma -(n+1)s]=\left[-{\displaystyle \frac{n(n+1)\epsilon }{2}},{\displaystyle \frac{n(n+1)\epsilon }{2}}\right],$$

which converges to $\left\{0\right\}$, the other eigenvalue of the limit matrix, as $\epsilon \to 0$.

The following Section studies Hankel TP matrices, whose eigenvalues are all nonnegative due to their total positivity ([18]).

4. Total Positivity and Hankel Matrices

In order to check whether a given matrix A is STP, let us recall that a characterization considerably reducing the number of minors to consider is given by Theorem 4.1 of [34], which only uses the initial minors defined in Section 2. This characterization improved an old characterization by Fekete in a paper [35] by Fekete and Polya that used the positivity of all minors $\det A\left[\alpha \right|\beta ]$ with $d\left(\alpha \right)=0=d\left(\beta \right)$.

Theorem 3.

An $n\times m$ matrix A is STP if, and only if, all its initial minors are positive.

For a Hankel matrix A, it is known that a characterization (cf. Theorem 4.4 of [4]) shows the equivalence of the strict total positivity of A with the positive definiteness of A and a symmetric submatrix of A.

Theorem 4.

An $(n+1)\times (n+1)$ Hankel matrix A is STP if, and only if, the symmetric matrices A and $A[1,\dots ,n|2,\dots ,n+1]$ are positive definite.

In the case of $(n+1)\times (n+1)$ Hankel matrices, the following result uses very few minors, reducing the characterization of strict total positivity to the positivity of $2n+1$ minors.

Theorem 5.

An $(n+1)\times (n+1)$ Hankel matrix A is STP if, and only if, its leading principal minors and the leading principal minors of $A[1,\dots ,n|2,\dots ,n+1]$ are positive.

Proof. 

Since an STP matrix has all its minors positive, it is sufficient to prove that the positivity of the leading principal minors of the statement implies that A is STP. As is well known, a symmetric matrix is positive definite if, and only if, all its leading principal minors are positive. Since matrices A and $A[1,\dots ,n|2,\dots ,n+1]$ are symmetric, both are positive definite. Now, the result follows from Theorem 1. □

The following result for positive Hankel matrices with first parameters $b_1\ge b_0$ provides a necessary condition for being TP₂. The condition is that the whole sequence of parameters is monotone increasing, extending the property satisfied by the two first parameters. We can see that it is a very simple condition to assure that the matrix is TP₂, which is a property that is very important in many applications (cf. [19,27]).

Proposition 3.

Let A be a positive Hankel matrix with parameters ${\left\{b_k\right\}}_{k=0}^{2n}$ and with $b_0\le b_1$. If A is TP₂, then $(0<b_0\le )b_1\le \cdots \le b_{2n}$.

Proof. 

Since A is TP₂, $0\le \det A[1,2]=b_0{b}_2-b_1^2$. Since $b_0>0$ and $b_1\ge b_0$, we deduce from the previous inequality that

$$b_2\ge {\displaystyle \frac{b_1^2}{b_0}}\ge {\displaystyle \frac{b_1{b}_0}{b_0}}=b_1.$$

Analogously, since $b_1>0$, $b_2\ge b_1$ and $\det A[1,2|2,3]\ge 0$, we can deduce that $b_3\ge b_2$. Continuing in a similar way, and using that $\det A[1,2|3,4]\ge 0,\dots ,\det A[1,2|n,n+1]\ge 0,$$\det A[2,3|n,n+1]\ge 0,\det A[3,4|n,n+1]\ge 0,\dots ,\det A[n-1,n|n,n+1]\ge 0$ and $\det A[n,n+1]\ge 0$, we prove that all parameters are positive and that $b_3\ge b_2,$$b_4\ge b_3,b_{n+1}\ge b_n,b_{n+2}\ge b_{n+1},b_{n+3}\ge b_{n+2},\dots ,b_{2n}\ge b_{2n-1}$. Hence, the result follows. □

We now present a sufficient condition for the strict total positivity of a positive Hankel matrix using a condition derived in [36], which, in turn, slightly improves a condition of [37] (see Corollary 2.17 of [4]). This sufficient condition only uses $2n-2$ inequalities for an $(n+1)\times (n+1)$ Hankel matrix. Since each inequality involves two products and one quotient, this condition only requires $4n-4$ products, $2n-2$ divisions and $2n-2$ comparisons, and it guarantees the positivity of all minors of the matrix.

Proposition 4.

If A is a positive Hankel matrix with parameters ${\left\{b_k\right\}}_{k=0}^{2n}$ satisfying $b_k\ge {\displaystyle \frac{4b_{k-1}^2}{b_{k-2}}}$ for all $k=2,3,\dots ,2n$, then A is STP.

Proof. 

Observe that, due to the form of A, the sufficient condition of Corollary 2.17 of [4] is of the form $b_k{b}_{k-2}\ge 4b_{k-1}^2$ for all $k=2,3,\dots ,2n$, which follows from our hypothesis. □

Hankel TP matrices are not as closely related with Toeplitz TP matrices as one might imagine. In fact, they have many different properties. However, there is a class of sign regular (SR) matrices that is very close to the class of Hankel TP matrices. Our following result relates Hankel TP matrices with that special class of Toeplitz SR matrices. For this purpose, we shall consider the $n\times n$ matrix P, which is the reverse of the identity matrix (i.e., it has 1 in entries $(1,n),(2,n-1),\dots ,(n,1)$ and 0 elsewhere).

We also need to recall the following result (see Theorem 3.1 of [18]), which claims that the product of SR matrices is an SR matrix.

Proposition 5.

Let A and B be two $n\times n$ SR matrices with signature ${\epsilon }^{\left(A\right)}=({\epsilon }_1^{\left(A\right)},\dots ,{\epsilon }_n^{\left(A\right)})$ and ${\epsilon }^{\left(B\right)}=({\epsilon }_1^{\left(B\right)},\dots ,{\epsilon }_n^{\left(B\right)})$, respectively. Then, the product $AB$ is a SR matrix with signature $({\epsilon }_1^{\left(A\right)}{\epsilon }_1^{\left(B\right)},\dots ,{\epsilon }_n^{\left(A\right)}{\epsilon }_n^{\left(B\right)})$. If, in addition, A is nonsingular and B is SSR, the $AB$ is SSR.

We can now give the result relating Hankel TP matrices with Toeplitz SR matrices with a special signature.

Proposition 6.

An $n\times n$ matrix A is a Hankel TP (respectively, STP) if, and only if, $AP$ is Toeplitz SR (respectively, SSR) with signature $\epsilon ={\left({\epsilon }_i\right)}_{1\le i\le n}$ given by ${\epsilon }_i={(-1)}^{\lfloor {\displaystyle \frac{i}{2}}\rfloor }$ for $i=1,\dots ,n$.

Proof. 

Let us observe that an $n\times n$ TP (respectively, STP) matrix is an SR (respectively, SSR) matrix whose signature is $\epsilon ={\left({\epsilon }_i\right)}_{1\le i\le n}$ with ${\epsilon }_i=1$ for all $1\le i\le n$ and that P is a nonsingular SR matrix whose signature is $\epsilon ={\left({\epsilon }_i\right)}_{1\le i\le n}$ with ${\epsilon }_i={(-1)}^{\lfloor {\displaystyle \frac{i}{2}}\rfloor }$ for $i=1,\dots ,n$, where by $\lfloor x\rfloor$, we denote the greatest integer less than or equal to x.

If A is Hankel, then $AP$ is Toeplitz and, if A is TP (respectively, STP), then by Proposition 5, we have that $AP$ is an SR (respectively, SSR) matrix with the signature of P.

For the converse, since $P^2$ is the identity matrix, observe that, if $AP$ is Toeplitz with the signature of P, then $\left(AP\right)P=A$ is Hankel and again by Proposition 5, we have that A is SR with the signature vector formed by 1’s, that is, A is TP. □

The importance of this result has two aspects. On the one hand, results on Toeplitz SR matrices with the signature given in Proposition 6 lead to corresponding results on the class of Hankel TP matrices. On the other hand, new results on Hankel TP matrices imply corresponding results on Toeplitz matrices with the signature given in Proposition 6. This last aspect is even more important than the other one because the properties of SR matrices are usually much more difficult to obtain than the properties of TP matrices. For instance, there is no result analogous to Theorem 3 for the class of SSR matrices. In fact, checking that a matrix is SSR involves a considerably greater number of minors (cf. [18,35]) than the number of minors used in Theorem 3. In contrast, we can see that only $2n+1$ are required to check whether a Toeplitz matrix is SSR with the signature given in Proposition 6 when we use Theorem 5.

5. Conclusions

Eigenvalue localization for positive Hankel matrices is analyzed. An interval containing all the eigenvalues of a positive Hankel matrix is presented. With an extra condition, we also provide two intervals with empty intersection that are contained in the previous one and such that they also contain all the eigenvalues. In fact, both intervals can be very useful for the eigenvalue localization of positive Hankel matrices because they can be very small. We have found a family of matrices illustrating that both intervals are necessary and that they can give a very sharp localization of the eigenvalues. Moreover, they can even give the exact eigenvalues.

Hankel TP and STP matrices were also studied. In the last case, they can be characterized in terms of the positivity of the $2n+1$ minors for an $(n+1)\times (n+1)$ matrix. A necessary condition for the total positivity of order 2 of positive Hankel matrices with the first parameter bounded above by the second parameter is proved. A condition assuring the strict total positivity of an $(n+1)\times (n+1)$ Hankel matrix using $2n-2$ inequalities is presented.

Finally, the relationship between Hankel TP matrices and Toeplitz SR matrices of a special signature is shown.

We now note some possible extensions of our work that can lead to future research. On the one hand, seeking to generalize our results and tools to classes of matrices that generalize Hankel matrices. On the other hand, obtaining new properties of SR matrices with the signature given in Proposition 6 by using the properties of Hankel TP matrices and Proposition 6.