Eigenvalue Localization for Symmetric Positive Toeplitz Matrices

Abstract

Given a real symmetric matrix, several inclusion and exclusion intervals containing its eigenvalues can be given. In this paper, for symmetric positive Toeplitz matrices, we provide an inclusion interval and, under an additional hypothesis, we also give two disjoint intervals contained in the previous one and containing all the eigenvalues. Examples are included, showing that these two intervals are necessary and that they can provide precise information on the localization of the eigenvalues. Sufficient conditions for positive definiteness are included. Necessary and sufficient conditions for the total positivity of symmetric positive Toeplitz matrices are presented. A characterization of symmetric totally positive circulant matrices is also obtained.

1. Introduction

The problem of obtaining regions of the complex plane containing all eigenvalues is an old subject that still leads to an important and active research topic (cf. [1,2,3,4]). If the matrix satisfies special properties, the eigenvalue localization problem has also special properties. For instance, if the matrix is symmetric, then all its eigenvalues are real. In this paper, we consider the eigenvalue localization problem for symmetric positive Toeplitz matrices, being aware that a huge body of literature exists starting from the early years of the past century, when Toeplitz matrices began to be associated to a generating function; see [5,6,7,8,9,10,11,12] and the references therein.

Toeplitz matrices arise in many applications of different fields, such as signal processing, physics, differential and integral equations or statistics (see [9,10] and references therein). Our approach to study the eigenvalue localization problem for these matrices is based on classical Gerschgorin circles as well as on an alternative approach presented in [13,14]. In contrast to Gerschgorin circles, which are sharper when the matrix is close to a diagonally dominant matrix, this alternative approach has a complementary behavior, in the sense that it is sharper when the matrix is far from a diagonally dominant matrix. In fact, it is sharper when the off-diagonal entries have close sizes.

Another topic considered in this paper is total positivity. This is also a field with more than a century of history and that is now receiving a lot of attention, leading to a very active research area. Totally positive matrices have many application to combinatorics, economics, approximation theory, finances, differential equations, statistics, mechanics, quantum groups, computer-aided geometric design, Markov chains, and biomathematics (cf. [15,16,17,18,19,20,21,22,23,24,25]). Moreover, the relationship of Toeplitz matrices with total positivity is also an old and important subject, where this paper also presents some new results.

The paper is organized in the following way. Section 2 presents the main classes of matrices used in this paper. In addition to the classes mentioned above, circulant matrices are defined. They form a subclass of Toeplitz matrices. Section 3 recalls some inclusion and exclusion eigenvalues regions obtained with the approaches commented previously, and it particularizes them for the case of symmetric positive matrices. In Section 4, we present an inclusion interval for the eigenvalues of these matrices and, under an additional hypothesis, it also provides two disjoint intervals contained in the previous one and containing all the eigenvalues. As we show with some examples, these two intervals are necessary, and they can provide sharp information on the localization of the eigenvalues. Some sufficient conditions for the positive definiteness of these matrices are also provided.

Section 5 discusses the relationship of Toeplitz matrices with total positivity and presents some necessary and sufficient conditions of total positivity for these matrices. It also characterizes when a symmetric circulant matrix is totally positive. Another related commented field is that of obtaining algorithms with high relative accuracy for linear algebra problems. In this field, totally positive matrices and Toeplitz matrices have a different behavior in general, although there are connections in some tridiagonal Toeplitz matrices. Section 6 revisits another interval for the eigenvalues localization by using generating functions of Toeplitz matrices and their powerful associated tools. Finally, Section 7 presents the main conclusions of the paper.

2. Some Classes of Matrices

This section is devoted to introducing some of the classes of matrices that will be used in this paper and to recall some of their properties, in particular, their eigenvalues.

In this paper, we say that a matrix $A={\left(a_{ij}\right)}_{1\le i,j\le n}$ is positive (respectively, non-negative) if $a_{ij}>0$ (respectively, $a_{ij}\ge 0$) for all $i,j$. We shall consider symmetric matrices whose eigenvalues are all real, and positive definite matrices whose eigenvalues are all positive.

A diagonal matrix with equal diagonal entries is a scalar matrix. A matrix where all the entries are the same number is called a constant matrix. A constant matrix with all entries equal to zero is called the null matrix. An $n\times n$ Toeplitz matrix $A={\left(a_{ij}\right)}_{1\le i,j\le n}$ is a real matrix such that all its diagonals are constant. These matrices can be defined through a sequence of $2n-1$ real numbers ${\left\{a_k\right\}}_{k=-n+1}^{n-1}$ with

$$a_{ij}:=a_{i-j},1\le i,j\le n,$$

(1)

which are called the parameters associated to the Toeplitz matrix A. Thus, a Toeplitz matrix A with parameters ${\left\{a_k\right\}}_{k=-n+1}^{n-1}$ is of the form

$$A=\left(\begin{array}{ccccc}{a}_0& a_1& \cdots & a_{n-2}& a_{n-1}\\ a_{-1}& a_0& \ddots & & a_{n-2}\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ ⋮& & \ddots & \ddots & a_1\\ a_{-n+1}& \cdots & \cdots & a_{-1}& a_0\end{array}\right).$$

If the Toeplitz matrix is symmetric, then it can be defined by (1) for $i\ge j$, and it only uses the n parameters ${\left\{a_k\right\}}_{k=0}^{n-1}$. Toeplitz matrices arise in many applications: in signal processing, in physics, in solutions to differential and integral equations, and in statistics.

A common special case of Toeplitz matrices is formed by the circulant matrices, where every row of the matrix is a right cyclic shift of the row above: the parameters satisfy $a_k=a_{-n+k}$ for all $k=1,\dots ,n-1$. Eigenvalues and eigenvectors of circulant matrices are known (cf. [26]). For symmetric circulant matrices, we have the extra condition $a_{n-i}=a_i$ for all i. Thus, they have the form

$$A=\left(\begin{array}{ccccc}{a}_0& a_1& \cdots & a_2& a_1\\ a_1& a_0& \ddots & & a_2\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ ⋮& & \ddots & \ddots & a_1\\ a_1& \cdots & \cdots & a_1& a_0\end{array}\right).$$

(2)

In addition to Toeplitz matrices, we consider in this paper another important class of matrices. A real matrix is totally positive (TP) if all its minors are non-negative (see [21,24,25]). These matrices are also called totally non-negative matrices, and they arise in many fields such as combinatorics, economics, approximation theory, finances, differential equations, statistics, mechanics, quantum groups, computer-aided geometric design, Markov chains, and biomathematics (cf. [15,16,17,18,19,20,21,22,23,24,25]). By Corollary 6.6 of [25], all eigenvalues of a TP matrix are non-negative numbers. When the matrix has all its minors as strictly positive, it is called strictly totally positive (STP). By Theorem 6.2 of [25], a STP matrix has all its eigenvalues as positive and distinct.

Finally, we present a weaker property than total positivity. A real matrix is TP₂ if all its $2\times 2$ submatrices have a non-negative determinant. Many properties and applications of TP₂ matrices can be seen in [21].

3. Eigenvalue Localization of Symmetric Positive Matrices

Given a real matrix $A={\left(a_{ij}\right)}_{1\le i,j\le n}$, several eigenvalue regions are obtained. First, let us mention Gerschgorin circles. For each $i=1,\dots ,n$, we denote by

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

the ith row Gerschgorin circle. Analogously, the column Gerschgorin circles can be defined. By Gerschgorin’s theorem, all eigenvalues belong to the union of all row Gerschgorin circles, and also to the union of all column Gerschgorin circles. An excellent book on this and related topics is [4].

For the real parts of the eigenvalues, alternative inclusion regions were obtained in [13]. Let us now recall them. For this purpose, let us define for each $i=1,\dots ,n$

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

(3)

and

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

(4)

Analogously, we define for each $j=1,\dots ,n$

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

(5)

and

$$c_j:=\left\{\begin{array}{cc}{c}_j^{+},\hfill & \mathrm{if}{a}_{jj}>0,\hfill \\ c_j^{-},\hfill & \mathrm{if}{a}_{jj}<0.\hfill \end{array}\right.$$

(6)

Number $r_i$ leads (see Proposition 3.3 of [13]) to a characterization of the class of $\overline{B}$-matrices, which is a class of matrices used in [13] to obtain eigenvalue localization regions known as $\overline{B}$-intervals and that will be recalled in the next result. Let us mention that some extensions of $\overline{B}$-matrices to tensors were obtained in [27,28,29,30].

By Theorem 3.5 (i) of [13] (applied to A and $A^T$), we have the following inclusion regions for the real eigenvalues of a real matrix A.

Theorem 1.

Let $A={\left(a_{ik}\right)}_{1\le i,k\le n}$ be a real matrix, and let $r_i^{+},r_i^{-}{c}_i^{+},c_i^{-}$ be as in (3) and (5). If λ is a real eigenvalue of A, then $\lambda \in {\bigcup }_{i=1}^n[a_{ii}-r_i^{+}-{\sum }_{k\ne i}|r_i^{+}-a_{ik}|,a_{ii}-r_i^{-}+{\sum }_{k\ne i}|r_i^{-}-a_{ik}\left|\right]$ and $\lambda \in {\bigcup }_{j=1}^n[a_{jj}-c_j^{+}-{\sum }_{k\ne j}|c_j^{+}-a_{kj}|,a_{jj}-c_j^{-}+{\sum }_{k\ne j}|c_j^{-}-a_{kj}\left|\right]$.

We shall call the intervals of the first set of Theorem 1 row $\overline{B}$-intervals. When we use $c_i^{+},c_i^{-}$ of (5) instead of (3), we call the corresponding intervals column $\overline{B}$-intervals.

Moreover, $\overline{B}$-intervals can be also used to localize the real parts of all eigenvalues of a real matrix as Theorem 4.3 (i) of [13] shows.

Theorem 2.

Let $A={\left(a_{ik}\right)}_{1\le i,k\le n}$ be a real matrix, and let $r_i^{+},r_i^{-},c_i^{+},c_i^{-}$ be as in (3) and (5). If λ is an eigenvalue of A, then $\mathit{Re}\left(\lambda \right)\in {\cup }_{i=1}^n[{\alpha }_i,{\beta }_i]$, where for each $i=1,\dots ,n$

$$\begin{array}{cc}\hfill & {\alpha }_i:=\min \{a_{ii}-r_i^{+}-\sum _{k\ne i}|r_i^{+}-a_{ik}|,a_{ii}-c_i^{+}-\sum _{k\ne i}|c_i^{+}-a_{ki}\left|\right\},\hfill \\ \hfill & {\beta }_i:=\max \{a_{ii}-r_i^{-}+\sum _{k\ne i}|r_i^{-}-a_{ik}|,a_{ii}-c_i^{-}+\sum _{k\ne i}|c_i^{-}-a_{ki}\left|\right\}.\hfill \end{array}$$

Since we are now interested in the localization of the eigenvalues of a symmetric positive matrix A, the row Gerschgorin circles reduce to the following intervals containing all eigenvalues of A: for each $i=1,\dots ,n$, the interval

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

(7)

As for the column Gerschgorin circles, for each $j=1,\dots ,n$, the interval

$$[a_{jj}-\sum _{k\ne j}{a}_{kj},a_{jj}+\sum _{k\ne j}{a}_{kj}].$$

(8)

The next result simplifies the previous notations when the matrix is symmetric positive and shows that, in this case, each of the previous row and column Gerschgorin intervals can only differ from the corresponding $\overline{B}$-interval in the left endpoints.

Proposition 1.

If $A={\left(a_{ik}\right)}_{1\le i,k\le n}$ is a symmetric positive matrix and $r_i^{+},r_i^{-},r_i,c_i^{+},c_i^{-},c_i$ are given in (3)–(6), then the eigenvalues of A belong to the intersection of the following two sets: the union of row Gerschgorin intervals and the union of row $\overline{B}$-intervals. For each $i=1,\dots ,n$, $r_i=r_i^{+}=c_i=c_i^{+}>0$, $r_i^{-}=c_i^{-}=0$, and the right endpoint of each row (respectively, column) Gerschgorin interval (7) (respectively, (8)) coincides with the right endpoint in the corresponding row (respectively, column) $\overline{B}$-interval.

Proof. 

Since A is symmetric, the row $\overline{B}$-intervals and row Gerschgorin intervals coincide with the corresponding column $\overline{B}$-intervals and column Gerschgorin intervals, and the first part of the result is a consequence of Gerschgorin’s theorem and of Theorem 1. Observe that, since A is positive, by (3) and (4), $r_i=r_i^{+}>0$ and $r_i^{-}=0$ for all $i=1,\dots ,n$, and, by the symmetry of the matrix, we also have $c_i=c_i^{+}=r_i>0$ and $c_i^{-}=0$ for all $i=1,\dots ,n$. The second part of this proposition follows from Theorem 1 and the properties $r_i^{-}=c_i^{-}=0$ for all $i=1,\dots ,n$. □

Until now, we only considered inclusion regions for the eigenvalues. We now recall an exclusion region for the real eigenvalues of a real matrix derived in [14]. Analogously to the class of $\overline{B}$-matrices used for the $\overline{B}$-intervals, in this case, we use the class of $\overline{C}$-matrices. The extension of $\overline{C}$-matrices to tensors was obtained in [31]. We first need some auxiliary notations. Given a real matrix $A={\left(a_{ik}\right)}_{1\le i,k\le n}$, for each $i=1,\dots ,n$, we define

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

(9)

Proposition 2.6 of [14] provides the following exclusion interval for the real eigenvalues of a real matrix.

Theorem 3.

Let $A={\left(a_{ik}\right)}_{1\le i,k\le n}$ be a real matrix, and let $s_i^{+},s_i^{-}$ be as in (9). If λ is a real eigenvalue of A, then

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

(10)

The interval of (10) is called the row exclusion interval.

Using $A^T$ instead of A, we can obtain an exclusion column interval, and a result analogous to Theorem 3 for the exclusion column interval also holds.

The following result simplifies the previous notations and the row exclusion interval when the matrix is symmetric positive.

Proposition 2.

If $A={\left(a_{ik}\right)}_{1\le i,k\le n}$ is a symmetric positive matrix and $s_i^{+},s_i^{-}$ are given in (9), then, for each $i=1,\dots ,n$, $s_i^{+}>0$ and $s_i^{-}=0$, and the eigenvalues of A cannot belong to the interval

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

(11)

Proof. 

Observe that since A is positive, by (9), $s_i^{+}>0$ and $s_i^{-}=0$ for all $i=1,\dots ,n$. The second part of this proposition follows from Theorem 3 and the properties $s_i^{-}=0$ for all $i=1,\dots ,n$. □

The next section considers the special case of symmetric positive Toeplitz matrices.

4. Inclusion Intervals for Symmetric Positive Toeplitz Matrices

Let us consider a symmetric positive Toeplitz matrix A with parameters ${\left\{a_k\right\}}_{k=0}^{n-1}$. We shall use the following notations:

$$m:=\underset{i>0}{\min }\left\{a_i\right\},M:=\underset{i>0}{\max }\left\{a_i\right\},$$

(12)

and the following notations for the minimal and maximal row sums of A:

$$\sigma :=\min \{\sum _{k=0}^{n-1}{a}_k,a_1+\sum _{k=0}^{n-2}{a}_k,\dots ,\sum _{k=n-1}^0{a}_k\},\Sigma :=\max \{\sum _{k=0}^{n-1}{a}_k,a_1+\sum _{k=0}^{n-2}{a}_k,\dots ,\sum _{k=n-1}^0{a}_k\}.$$

(13)

The following result first provides a simple eigenvalue inclusion interval for all eigenvalues of a symmetric positive Toeplitz matrix, and, under an additional condition, it also provides two disjoint intervals contained in the previous one and containing all the eigenvalues. As we shall see in Remark 1 for some examples, these two intervals are necessary, and they can provide sharp information on the localization of the eigenvalues.

Theorem 4.

Let $A={\left(a_{ij}\right)}_{1\le i,j\le n}$ be a symmetric positive Toeplitz matrix with parameters ${\left\{a_k\right\}}_{k=0}^{n-1}$ and let $m,M,\sigma ,\Sigma$ be given by (12) and (13). Then, all eigenvalues of A belong to the interval

$$[\max \{\sigma -nM,2a_0-\Sigma \},\Sigma ].$$

(14)

Moreover, if in addition $\Sigma -nm<\sigma$, then all eigenvalues of A belong to the following union of two disjoint intervals:

$$[\max \{\sigma -nM,2a_0-\Sigma \},\Sigma -nm]\cup [\sigma ,\Sigma ].$$

(15)

Proof. 

By Proposition 1, all eigenvalues of A belong to the intersection of the union of row Gerschgorin intervals and the union of row $\overline{B}$-intervals, and the right endpoint of each row $\overline{B}$-interval coincides with the the right endpoint of the corresponding row Gerschgorin interval. Since A is positive, right endpoints of all row Gerschgorin intervals are given by the corresponding row sums, and so they are bounded above by $\Sigma$. Again by the positivity of A, all left endpoints of all row Gerschgorin intervals are given by $2a_0$ minus the sum of the entries of the corresponding row, and so they are bounded below by $2a_0-\Sigma$. By Theorem 1 and the positivity of A, all left endpoints of all row $\overline{B}$-intervals are given by the correspond row sum minus $n{r}_i^{+}$ and since the row sum is bounded below by $\sigma$ and $r_i^{+}\le M$ for all $i=1,\dots ,n$, we conclude that all left endpoints of all row $\overline{B}$-intervals are bounded below by $\sigma -nM$.

Let us now prove the second part of the result so that we also assume that $\Sigma -nm<\sigma$. If $\lambda$ is an eigenvalue of A, then by Proposition 2,

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

(16)

Since $m\le s_i^{+}$ for all $i=1,\dots ,n$, ${\sum }_{j=1}^n{a}_{ij}-n{s}_i^{+}\le {\sum }_{j=1}^n{a}_{ij}-nm$ for all $i=1,\dots ,n$. Therefore,

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

Thus, (16) becomes

$$\lambda \notin \left(\Sigma -nm,\sigma \right).$$

(17)

By our hypothesis, the exclusion interval of (17) is nonempty and the result follows. □

Observe that symmetric positive circulant matrices obviously satisfy all hypotheses of Theorem 4. In particular, $\Sigma -nm<\sigma$ because, for these matrices, $\Sigma =\sigma$. The next remark will show examples of these matrices for which both disjoint intervals of (15) are necessary and, in fact, they provide exactly the eigenvalues of the matrices. In this sense, Theorem 4 cannot be improved.

Remark 1.

Let us consider the $n\times n$ ($n\ge 2$) symmetric Toeplitz matrix A with n positive parameters $\{a_0,a_1,\dots ,a_1\}$, that is,

$$A=\left(\begin{array}{ccccc}{a}_0& a_1& \cdots & \cdots & a_1\\ a_1& a_0& \ddots & & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ ⋮& & \ddots & \ddots & a_1\\ a_1& \cdots & \cdots & a_1& a_0\end{array}\right).$$

Clearly, this matrix has the eigenvalues $a_0+(n-1)a_1$ associated to the eigenvector $(1,\dots ,1)$ and the eigenvalue $a_0-a_1$ with multiplicity $n-1$. A satisfies all hypotheses of Theorem 4. Its two disjoint intervals of (15) are

$$[\max \{\sigma -nM,2a_0-\Sigma \},\Sigma -nm]=[a_0-a_1,a_0-a_1]=\{a_0-a_1\}$$

and

$$[\sigma ,\Sigma ]=[a_0+(n-1)a_1,a_0+(n-1)a_1]=\{a_0+(n-1)a_1\},$$

which exactly provide the two eigenvalues of A.

As a consequence of the first part of Theorem 4, we can also obtain a sufficient condition for the positive definiteness of a symmetric positive Toeplitz matrix.

Corollary 1.

Let $A={\left(a_{ij}\right)}_{1\le i,j\le n}$ be a symmetric positive Toeplitz matrix and let $M,\sigma ,\Sigma$ be given by (12) and (13). If A satisfies that

$$\max \{\sigma -nM,2a_0-\Sigma \}>0,$$

then A is positive definite.

Proof. 

By Theorem 4, all eigenvalues are greater than or equal to the left endpoint of the interval (14), which is positive by our hypothesis. □

Let us illustrate the application of Corollary 1 with two extreme matrices.

Example 1.

First, let us consider the $n\times n$ ($n\ge 2$) symmetric Toeplitz matrix B with n positive parameters $\{a+\epsilon ,a,\dots ,a\}$ with $a,\epsilon >0$. In this case, the maximum of Corollary 1 is $\sigma -nM=na+\epsilon -na=\epsilon >0$. The second example is given by the $n\times n$ ($n\ge 4$) symmetric Toeplitz matrix A with n positive parameters $\{2+(n-2)\epsilon ,1,\epsilon ,\dots ,\epsilon ,1\}$ with $\epsilon >0$. In this case, the maximum of Corollary 1 is $2a_0-\Sigma =4+2(n-2)\epsilon -(2+(n-2)\epsilon )-(n-3)\epsilon -2=\epsilon >0$.

The previous examples are only illustrative. In fact, since the matrices are circulant, the eigenvalues and the eigenvectors are explicitly known. The following example shows a family of matrices depending on $\epsilon >0$ for which the two intervals of (15) converge to the two eigenvalues of the limit matrix of this family as $\epsilon \to 0$.

Example 2.

Let us consider the family of symmetric Toeplitz matrices $A_{\epsilon }$ with n positive parameters $\{1,1-\epsilon ,\dots ,1-(n-1\left)\epsilon \right\}$ with $0<\epsilon <{\displaystyle \frac{1}{n}}$, $n>2$. As $\epsilon \to 0$, matrices $A_{\epsilon }$ tend to the matrix with all entries 1, whose eigernvalues are 0 and n. Observe that, for matrices $A_{\epsilon }$, σ and Σ given by (13) satisfy $\sigma =\Sigma =n-{\displaystyle \frac{n(n-1)\epsilon }{2}}$, which is an eigenvalue for all matrices $A_{\epsilon }$ associated to the eigenvector ${(1,\dots ,1)}^T$. So, the second interval $[\sigma ,\Sigma ]$ of Theorem 4 reduces to this eigenvalue. On the other hand, one can check for all matrices $A_{\epsilon }$ that $\max \{\sigma -nM,2a_0-\Sigma \}=\sigma -nM$ and so the first interval of Theorem 4 is

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

which tends to $\left\{0\right\}$ (the other eigenvalue of the limit matrix) as $\epsilon \to 0$.

The next section considers TP Toeplitz matrices, which have non-negative eigenvalues as recalled in the Introduction.

5. Total Positivity and Toeplitz Matrices

Definitions and some properties of TP matrices are recalled in Section 2. There are old and deep connections between Toeplitz matrices and total positivity.

Let us start by recalling the connection with the study of generating functions for TP infinite Toeplitz matrices (see Section 7 III of [25] and Section 4.7 of [24]). Given a bi-infinite sequence

$$\dots ,a_{-2},a_{-1},a_0,a_1,a_2,\dots$$

the matrix $A={\left(a_{j-i}\right)}_{i,j=1}^{\infty }$ is called its associated Toeplitz matrix, and the function $f\left(z\right)={\sum }_{n=-\infty }^{\infty }{a}_n{z}^n$ its generating function. The study and characterization of the total positivity Toeplitz matrix in terms of the generating functions was performed in [32,33,34,35,36]. We now recall the particular case of a one-sided infinite sequence

$$a_0,a_1,a_2,\dots$$

where we suppose that $a_{-i}=0$ for $i=1,2,\dots ,$ in the previous definition. Assuming the normalization $a_0=1$, the previous sequence gives rise to a TP Toeplitz matrix $A={\left(a_{j-i}\right)}_{i,j=1}^{\infty }$ if and only if its corresponding generating function ${\sum }_{n=0}^{\infty }{a}_n{z}^n$ has the form

$$e^{\gamma z}{\displaystyle \frac{{\Pi }_{k=1}^{\infty }(1+{\alpha }_{k}z)}{{\Pi }_{k=1}^{\infty }(1-{\beta }_{k}z)}},$$

where $\gamma ,{\alpha }_k,{\beta }_k\ge 0$, and ${\sum }_{k=1}^{\infty }({\alpha }_k+{\beta }_k)<\infty$. Sequences leading to TP Toeplitz matrices were recalled in Chapter VIII of [21] Polya frequency sequences (PF-sequences), where many examples were presented. Examples of PF-sequences arising in combinatorics can be seen in [37]. Let us also recall that results concerning the total positivity of the Hadamard product of TP Toeplitz matrices can be seen in [38].

Now we come back to finite symmetric positive Toeplitz matrices, and we shall present necessary and sufficient conditions related to total positivity. We first give a necessary condition for TP₂ in a very simple way: the parameters form a monotone decreasing sequence. If a stronger condition is required, then we obtain a sufficient condition for strict total positivity. This last sufficient condition uses a corresponding condition obtained in [39], which slightly refined a previous condition of [40] (see Corollary 2.17 of [24]).

Proposition 3.

Let A be a symmetric positive Toeplitz matrix with parameters ${\left\{a_k\right\}}_{k=0}^{n-1}$. If A is TP₂, then $a_0\ge a_1\ge \cdots \ge a_{n-1}(>0)$. If $a_1\le {\displaystyle \frac{a_0}{2}}$ and $a_{k+1}\le {\displaystyle \frac{a_k^2}{4a_{k-1}}}$ for all $k=1,2,\dots ,n-2$, then A is STP.

Proof. 

Let us first assume that A is TP₂. If we consider the submatrix formed by rows 1, 2 and columns 1, 2, we obtain $a_0^2\ge a_1^2$ and so the positivity of the parameters implies $a_0\ge a_1$. With the submatrix formed by rows 1, 2 and columns 2, 3, we obtain $a_1^2\ge a_0{a}_2\ge a_1{a}_2$ and so the positivity of the parameters implies $a_1\ge a_2$. Continuing in an analogous way, we obtain $a_0\ge a_1\ge \cdots \ge a_{n-1}(>0)$.

For the second part of the result, let us assume that $a_1\le {\displaystyle \frac{a_0}{2}}$ and $a_{k+1}\le {\displaystyle \frac{a_k^2}{4a_{k-1}}}$ for all $k=2,\dots ,n-2$. By applying Corollary 2.17 of [24] to A, for its strict total positivity, it is sufficient to prove that $a_0^2\ge 4a_1$, which holds by our hypothesis, and that $a_k^2\ge 4a_{k-1}{a}_{k+1}$ for all $1\le k\le n-2$, which also holds by our hypothesis. □

A $2\times 2$ symmetric positive Toeplitz matrix is TP if and only if its associated parameters $\{a_0,a_1\}$ satisfy $a_0\ge a_1$. Let us now consider symmetric circulant matrices of the form (5). In this case, $n\times n$ ($n\ge 3$) TP symmetric circulant matrices are characterized in the following result.

Theorem 5.

An $n\times n$ ($n\ge 3$) symmetric circulant matrix A with parameters ${\left\{a_k\right\}}_{k=0}^{n-1}$ is TP if and only if it is either a non-negative constant matrix or a scalar matrix with a positive diagonal.

Proof. 

Trivially, any non-negative constant matrix is TP and any scalar matrix with positive diagonal is TP. Let us see the converse, and so we assume that A is a symmetric TP circulant matrix.

Since A is a circulant, it is in particular a Toeplitz matrix, and, since it is TP, it is also TP₂. If it is positive, then by Proposition 3, the parameters ${\left\{a_k\right\}}_{k=0}^{n-1}$ satisfy $a_0\ge a_1\ge \cdots \ge a_{n-1}$, which also holds when A is the null matrix. Since A is symmetric circulant, it also satisfies that $a_{n-i}=a_i$ for all i. Then, we have $a_0\ge a_1\ge \cdots \ge a_1$ and so, $a_0\ge a_1=a_2=\cdots =a_{n-1}\ge 0$. The non-negative determinant of the submatrix formed by rows 2, 3 and columns 1, 2 is given by $a_1^2-a_0{a}_1\le 0$, which implies that either $a_0=a_1$ (that is, A is a non-negative constant matrix) or $a_1=0$ (that is, A is a scalar matrix with a positive diagonal). □

We finish this section by commenting on the problem of finding algorithms with high relative accuracy (cf. [41,42,43]. For some subclasses of TP matrices (cf. [44,45,46,47]), algorithms with high relative accuracy have been found for solving linear algebra problems, such as the computation of the inverse, of all singular values, of all eigenvalues, and the solution of some linear systems. This can be performed by starting from an accurate bidiagonal decomposition of a TP matrix and using the algorithms presented in [48] (and in [49] for the inverse). In contrast, accurate linear algebra for the problem of calculating determinants or minors is impossible on the class of Toeplitz matrices (see corollaries 3.43 and 3.45 of [41]). However, for special classes of Toeplitz matrices, such as some tridagonal Toeplitz matrices, algorithms with high relative accuracy for some linear algebra problems were obtained in [50].

6. An Interval Using Generating Functions of Toeplitz Matrices

We already commented in Section 2 that eigenvalues and eigenvectors of circulant matrices are known. In fact, many spectral properties of Toeplitz matrices associated to a generating function (see Section 5) are known. Following p. 95 of [11], a Toeplitz matrix A given by (1) with parameters ${\left\{a_k\right\}}_{k=-n+1}^{n-1}$ can be written as

$$A=\sum _{k=-(n-1)}^{n-1}{a}_k{J}_n^{\left(k\right)},$$

where $J_n^{\left(k\right)}$ is the $n\times n$ matrix whose $(i,j)$th entry equals 1 if $i-j=k$ and 0 otherwise.

Given a function $f:[-\pi ,\pi ]\to bfC$ belonging to $L^1\left([-\pi ,\pi ]\right)$, its Fourier coefficients are given by

$$f_k={\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }f\left(\theta \right)e^{-ik\theta }\mathbf{d}\theta ,k\in \mathbf{Z},$$

where $i^2=-1$.

The nth Toeplitz matrix associated with f is defined as

$$T_n\left(f\right)=\sum _{k=-(n-1)}^{n-1}{f}_k{J}_n^{\left(k\right)}.$$

The sequence of Toeplitz matrices ${\left\{T_n\left(f\right)\right\}}_n$ is called the Toeplitz sequence generated by f, and f is called the generating function of ${\left\{T_n\left(f\right)\right\}}_n$. The generating function can be periodically extended on the real line and, after the change of variable $z=e^{i\theta }$, is the generating function given in Section 5. As shown in Section 6.2 of [11], if f is real-valued, then $T_n\left(f\right)$ is Hermitian for every n, and all its eigenvalues are real.

As shown in Theorem 6.1 of [11], taking $\tilde{m}$ as the essential infimum of f and $\tilde{M}$ as its essential supremum, then the interval $(\tilde{m},\tilde{M})$ contains all eigenvalues of $T_n\left(f\right)$ for all n. Furthermore, if $\alpha$ is the maximal order of the zeros of $f-\tilde{m}$ and $\beta$ is the maximal order of the zeros of $\tilde{M}-f$, then the minimal eigenvalue of $T_n\left(f\right)$ converges monotonically from above to $\tilde{m}$ with a convergence speed asymptotic to $1/n^{\alpha }$, and the maximal eigenvalue of $T_n\left(f\right)$ converges monotonically from below to $\tilde{M}$ with a convergence speed asymptotic to $1/n^{\beta }$; see [6,11,12] and the references therein.

These results can be applied to specific examples of real symmetric Toeplitz matrices with non-negative entries and arising in concrete applications. For instance, if one takes $g_2\left(\theta \right)={(2-2\cos \left(\theta \right))}^2$ with stencil $(a_{-2},a_{-1},a_0,a_1,a_2)=(1,4,6,4,1)$ (and with all the other Fourier coefficients equal to zero), then all the eigenvalues of $T_n\left(g_2\right)$ belong to $(0,16)=(\tilde{m},\tilde{M})$, and the minimal eigenvalue is equal to $c_1/n^4+o(1/n^4)$ and the maximal eigenvalue is equal to $16-c_2/n^4+o(1/n^4)$, with $c_1,c_2>0$ and where the value of $c_1,c_2$ can be explicitly identified; see [6,7] and the references therein. This example comes from applications because $T_n\left(g_2\right)$ represents a canonical Finite Difference discretization of ${\displaystyle \frac{d^4}{d{x}^4}}$ on $(0,1)$ with equispaced grid and proper homogeneous boundary conditions. Since $T_n\left(g_2\right)$ has zero entries for $n>3$, we cannot use our results from the previous sections. However, for $n=3$, it is a positive Toeplitz matrix and the intervals obtained in (15) are $[-1,11]\cup [11,14]=[-1,14]$.

In general, if one considers $g_k\left(\theta \right)={(2-2\cos \left(\theta \right))}^k$, with k as a positive integer, then one ends up again with real symmetric Toeplitz matrices with non-negative entries and discretizing ${\displaystyle \frac{d^{2k}}{d{x}^{2k}}}$ on $(0,1)$ with equispaced grid and proper homogeneous boundary conditions, via canonical Finite Differences. The same very beautiful localization and asymptotic results can be obtained. These results go beyond the case of real Fourier coefficients, beyond the Hermitian setting, and beyond the case of zeros of integer orders like in the approximation of fractional operators; see [5] and the references therein including [6,7]. And for the banded case, also asymptotic expansions of the eigenvalues can be found; see [8].

7. Conclusions

This paper studies the eigenvalue localization for a symmetric positive Toeplitz matrix. An inclusion interval for its eigenvalues is provided. With one additional restriction, we also give two disjoint intervals contained in the previous one and containing all the eigenvalues. We include examples showing that these two intervals are necessary, and they can provide sharp information on the localization of the eigenvalues. In fact, they can even provide exactly the eigenvalues of the matrices. Sufficient conditions to check if a given symmetric positive Toeplitz matrix is positive definite are provided. Illustrative examples are included.

The relationship between TP matrices and Toeplitz matrices is also analyzed. A necessary condition for the total positivity of a symmetric positive Toeplitz matrix in terms of its parameters is obtained. A sufficient condition for its strict total positivity is also presented. A characterization of symmetric totally positive circulant matrices is also obtained. They are formed by non-negative constant matrices and scalar matrices with positive diagonal.

Finally, the tools associated with generating functions and mentioned in Section 6 and the references therein are very general, and also provide an eigenvalue inclusion interval and apply to any kind of real symmetric (and also complex Hermitian, or non-Hermitian) Toeplitz matrices.