In this article, we express the eigenvalues of real antitridiagonal Hankel matrices as the zeros of given rational functions. We still derive eigenvectors for these structured matrices at the expense of prescribed eigenvalues.
Recently, some authors have computed the eigenvalues and eigenvectors for a sort of Hankel matrices, thus obtaining its eigendecomposition (see [7,8,12,15,16, 17,18], among others). Moreover, the spectral analysis for a type of persymmetric antitridiagonal 2-Hankel matrices was also undertaken (see ). In contrast to the tridiagonal Toeplitz case where the eigenvalues and eigenvectors are well known (see, for instance, ), a possible closed-form expression for the eigenvalues and eigenvectors of general antitridiagonal Hankel matrices is yet to be found. In this quest, da Fonseca  gave a step forward by expressing explicitly the eigenvalues of odd order antitridiagonal Hankel matrices having null northeast-to-southwest diagonal. It must also be emphasized that, although antibidiagonal matrices or antitridiagonal matrices with null antidiagonal can be treated as matrices having tridiagonal structure in what concerns to spectral purposes or powers computation (see [3,6]), general antitridiagonal matrices, and particularly those considered here, cannot be reduced to tridiagonal ones (see Remark 3 of ).
The aim of this short note is to give a contribution for that research. Specifically, we shall present an eigenvalue localization tool for real antitridiagonal Hankel matrices, providing also associated eigenvectors. To achieve our purpose, we shall use eigendecompositions of anticirculant matrices available in  to ensure a decomposition for the matrices under study at first and results concerning to the sum of (rank one) matrices in a final stage to obtain all formulae.
The rational functions exhibited in this article to locate eigenvalues of real antitridiagonal Hankel matrices, as well as the expressions for its eigenvectors, are given in the explicit form, which, on the one hand, can be easily implemented in computer algebra systems, and, on the other hand, are useful for further theoretical investigations in this subject.
2 Main results
Let be a positive integer and consider the following antitridiagonal Hankel matrix
where , and are real numbers. Throughout, we shall set
where denotes the imaginary unit.
2.1 Eigenvalue localization for
Our first statement is an eigenvalue localization theorem for matrices of the form (2.1).
Let be a positive integer, , and be real numbers, and, is given by (2.2),
(a) If is odd then the eigenvalues of that are not of the form , , are precisely the zeros of the function
Moreover, if are the eigenvalues of and , , are arranged in nondecreasing order as , then
(b) If is even, then the eigenvalues of that are not of the form , , , are precisely the zeros of the function:
Moreover, if are the eigenvalues of and , , , are arranged in a nondecreasing order as , then
2.2 Eigenvectors of
Owning the eigenvalues of in (2.1), we are able to determine the corresponding eigenvectors.
Let be an integer, , and be real numbers such that , and , ( ), , and be given by (2.3a), (2.3b), (2.3c), and (2.3d), respectively.
(a) If is odd, the zeros of (2.4a) are not of the form , , , and , then
is an eigenvector of associated with , .
(b) If is even, the zeros of (2.5a) are not of the form , , , , and , then
is an eigenvector of associated with , .
3 Lemmas and proofs
Let be a positive integer. Consider the following real anticirculant matrix
and the unitary discrete Fourier transform matrix , that is, the matrix defined by
where is given by (2.2). Our first auxiliary result is an orthogonal decomposition for (3.1). We shall denote by the conjugate transpose of any complex matrix.
Let be a positive integer, , and be real numbers, and , , and , be given by (2.2), (2.3a), (2.3b), respectively.
(a) If is odd be, then
where is the orthogonal matrix defined by
(b) If is even, then
where is the orthogonal matrix whose entries are given by
Let be a positive odd integer. According to Theorem 3.6 of , we have
with given by (3.2) and the following matrix
Note that the first column of has all components equal to ; their next columns are
for each , and the last ones are
for . Since
and we obtain that the entries of are given by (3.3b), which leads to (3.3a). Supposing a positive even integer , Theorem 3.7 in  ensures
with given by (3.2) and by the matrix
The first column of has all its components equal to . The next columns are given by (3.5) for , and the last ones are defined by (3.6) for each ; the th column of is
From identities (3.7) and (3.8), we obtain (3.4a). The proof is completed.□
The following statement is a decomposition for the matrices and plays a central role in the main results.
Let be a positive integer, , and bereal numbers, and , , , be given by (2.2), (2.3a), and (2.3b), respectively.
We only prove (a) since (b) can be proven in the same way. Consider a positive odd integer and the following matrices
From Lemma 1,
where is the matrix defined by (3.3b), is the matrix (3.1), and is the first row of , i.e.,
and is the last row of ,
Proof of Theorem 1
Consider a positive odd integer , given by (3.9), and , , , . According to Lemma 2, it should be noted that the matrix and
share the same eigenvalues. Let us adopt the notations of  by denoting the collection of all -element subsets of written in the increasing order; in addition, for any rectangular matrix , we shall indicate by the minor determined by the subsets and . Setting
we have from Theorem 1 of  that is an eigenvalue of (3.11) if and only if
provided that is not an eigenvalue of . Since
we obtain (2.4a). Let be the eigenvalues of and be arranged in nondecreasing order by some bijection defined in . Thus,
for each (see , page 242). By using Miller’s formula for the determinant of the sum of matrices (see , page 70), we can compute the characteristic polynomial of ,