Analytical solutions to some generalized and polynomial eigenvalue problems


 It is well-known that the finite difference discretization of the Laplacian eigenvalue problem −Δu = λu leads to a matrix eigenvalue problem (EVP) Ax =λx where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in a recent work of Strang and MacNamara. We generalize the results and develop analytical solutions to certain generalized matrix eigenvalue problems (GEVPs) Ax = λBx which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.


Introduction
It is well-known that the following tridiagonal Toeplitz matrix has analytical eigenpairs (λ j , x j ) with x j = (x j, , · · · , x j,n ) T where λ j = − cos(jπh), x j,k = c sin(jπkh), h = n + , j, k = , , · · · , n with ≠ c ∈ R (we assume that c is a nonzero constant throughout the paper); see, for example, [20, p. 514] for a general case of tridiagonal Toeplitz matrix. Following the constructive technique in [20, p. 515] that nds the analytical solutions to the matrix eigenvalue problem (EVP) Ax = λx, one can derive analytical solutions to the generalized matrix eigenvalue problem (GEVP) Ax = λBx where B is an invertible tridiagonal Toeplitz matrix. For example, let then, the GEVP Ax = λBx has analytical eigenpairs (λ j , x j ) with x j = (x j, , · · · , x j,n ) T where (see [16,Sec. 4] for a scaled case; A is scaled by /h while B is scaled by h) λ j = − + + cos(jπh) , x j,k = c sin(jπkh), h = n + , j, k = , , · · · , n.
These matrices or their scaled (by constants) versions arise from various applications. For example, the matrix A arises from the discrete discretizations of the 1D Laplace operator by the nite di erence method (FDM, cf., [23,25], scaled by /h ) or the spectral element method (SEM, cf., [22], scaled by /h). The work [27] showed that functions of the (tridiagonal) matrices that arise from nite di erence discretization of the Laplacian with di erent boundary conditions are Toeplitz-plus-Hankel. Similar analytical results for tridiagonal matrices from nite di erence discretization are obtained in [4]. The matrix B (and also A) arises from the discrete discretization by the nite element method (FEM, cf., [5,17,26], scaled by h).
In general, it is di cult to nd analytical solutions to the EVPs. The work by Losonczi [19] gave analytical eigenpairs to the EVPs for some symmetric and tridiagonal matrices. A new proof of these solutions was given by da Fonseca [6]. We refer to the recent work [7] for a survey on the analytical eigenpairs of tridiagonal matrices. For pentadiagonal and heptadiagonal matricies, nding analytical eigenpairs becomes more di cult. The work [2] derived asymptotical results of eigenvalues for pentadiagonal symmetric Toeplitz matrices. Some spectral properties were found in [14] for some pentadiagonal symmetric matrices. A recent work by Anđelić and da Fonseca [1] presents some determinantal considerations for pentadiagonal matrices. The work [24] gave analytical eigenvalues (as the zeros of some complicated functions) for heptadiagonal symmetric Toeplitz matrices.
To the best of the author's knowledge, there are no widely-known articles in literature which address the issue of nding analytical solutions to either the EVPs with more general matrices (other than the ones mentioned above) or the GEVPs. The articles [3,15,16] present analytical solutions (somehow implicitly) to GEVPs for some tridiagonal and/or pentadiagonal matrices that arise from the isogeometric analysis (IGA) of the 1D Laplace operator. For heptadiagonal and more general matrices, no analytical solutions exist and the numerical approximations in [3,8,15] can be considered as asymptotic results for certain structured matrices arising from the numerical discretizations of the di erential operators.
In this paper, we present analytical solutions to GEVPs for mainly two classes of matrices. The rst class is the Toeplitz-plus-Hankel matrices while the second class is the corner-overlapped block-diagonal matrices. The main insights for the Toeplitz-plus-Hankel matrices are from the numerical spectral approximation techniques where a particular solution form such as, the well-known Bloch wave form, is sought. For the corner-overlapped block-diagonal matrices, we propose to decompose the original problem into a lower twoblock matrix problem where one of the blocks is a quadratic eigenvalue problem (QEVP). We solve the QEVP by rewriting the problem and applying the analytical results from the tridiagonal Toeplitz matrices. Generalization of nding solutions to polynomial eigenvalue problem (PEVP) is also given. Additionally, Denton, Parke, Tao, and Zhang in a recent work [13] rediscovered and coined the eigenvector-eigenvalue identity for certain EVPs. We generalize this identity for the GEVPs. Based on these identities, we derive some interesting trigonometric identities.
The rest of the article is organized as follows. Section 2 presents the main results, i.e., the analytical solutions to the two classes of matrices. Several examples are given and discussed. Section 3 generalizes the eigenvector-eigenvalue identity and derives some trigonometric identities. Potential applications to the design of better numerical discretization methods for partial di erential equations and other concluding remarks are presented in Section 4.

Main results . Notation and problem statement
Throughout the paper, we denote matrices and vectors by uppercase and lowercase letters, respectively. In particular, let A ∈ C n×n be a square matrix with entries denoted as A jk , j, k = , · · · , n, and x ∈ C n be a vector with entries denoted as x j , j = , · · · , n, in the complex eld. We denote by A (α,m) a matrix that the entries depend on a sequence of parameters α = (α , α , · · · , αm). The superscript · (α,m) is omitted when the context is clear. We denote by T (α,m) = (T (α,m) j,k ) ∈ C n×n a symmetric and diagonal-structured Toeplitz matrix with entries where m ≤ n − speci es the matrix bandwidth. Explicitly, this matrix can be written as where the empty spots are zeros. For a Hankel matrix, it appears to be di erent in each of the cases we consider in the paper and we de ne it at its occurrence in the context. For matrices A, B ∈ C n×n , the EVP is to nd the eigenpairs (λ ∈ C, x ∈ C n ) such that while the GEVP is to nd the eigenpairs (λ ∈ C, x ∈ C n ) such that Throughout the paper, we focus on GEVPs and since the analytical eigenpairs for GEVPs with small dimensions are relatively easy to nd, we assume that the dimension n is large enough for the generalization of matrices to make sense. For simplicity, we slightly abuse the notation such as A for a matrix and (λ, x) for an eigenpair. Once analytic eigenpairs for a GEVP are found, eigenpairs for EVP Ax = λx follows naturally by setting B as an identity matrix.

. Toeplitz-plus-Hankel matrices
In this section, we present analytical solutions to certain Toeplitz-plus-Hankel matrices. The main insights are from the proof in nding analytical solutions to a tridiagonal Toeplitz matrix in [20, p. 514] and the numerical spectral approximations of the Laplace operator (cf., [3,15,16,27]). The main idea is to seek eigenvectors in a particular form such as the Bloch wave form e a+ιb , where ι = − as in [16,20] and sinusoidal form as in [15]. Our main contribution is the generalization of these results to GEVPs, especially, with larger matrix bandwidths.
We denote by H (α,m) = (H (α,m) j,k ) ∈ C n×n a Hankel matrix with entries which can be explicitly written as where the entries at the bottom-right corner are de ned such that the matrix is persymmetric. Herein, a persymmetric matrix is de ned as a square matrix which is symmetric with respect to the anti-diagonal. With this in mind, we give the following result. Proof. Following [20, p. 515], one seeks eigenvectors of the form c sin(jπkh). Using the trigonometric identity sin(ϕ ± ψ) = sin(ϕ) cos(ψ) ± cos(ϕ) sin(ψ), one can verify that each row of the GEVP (2.3), n k= A ik x j,k = λ n k= B ik x j,k , i = , · · · , n, reduces to α + m l= α l cos(ljπh) = λ β + m l= β l cos(ljπh) , which is independent of the row number i. Thus, the eigenpairs (λ j , x j ) given in (2.5) satis es (2.3). The GEVP has at most n eigenpairs and the n eigenvectors are linearly independent. This completes the proof.
We remark that all the matrices de ned above are symmetric and persymmetric. For m = and n ≥ , the where the Hankel matrix H (ξ ,m) modi es the matrix at the rst and last few rows. For m = , both A and B are tridiagonal Toeplitz matrices. For m ≥ , both A and B are Toeplitz-plus-Hankel matrices. This result generalizes the nite-di erence matrix results in Strang and MacNamara in [27].
We present the following example. Let m = , α = , α = − / , α = − / , β = / , β = / , β = / . Then, we have the following matrices The eigenpairs of the GEVP (2.3) with these matrices are If we scale the matrix A by /h while the matrix B by h, then the new system is a GEVP that arises from the IGA approximation of the Laplacian eigenvalue problem −∆u = λu on [ , ]; see, for example, [15, eqns. 118-123] (the rst two rows have slightly di erent entries A = / , one can shift the phase by a half and seek solutions of the form sin jπ(k − )h . As a consequence, we have the following result.
Herein, as an example, with m = and n ≥ , the matrix The proof can be established similarly and we omit it for brevity. An example is the GEVP with the matrices in [15, eqns. 118-123]. These matrices satisfy the assumption of the above theorem and the analytical solutions are given by (2.8) with a scale n to the eigenvalues. The eigenvectors of Theorems 2.1 and 2.2 correspond to the solutions of the Laplacian eigenvalue problem on the unit interval. When m ≥ , the results in Theorems 2.1 and 2.2 provide an insight to remove the outliers in high-order IGA approximations. We refer to [15,16] for the outlier behavior in the approximate spectrum and to [9,10] for their eliminations. Similarly, we de ne a Hankel matrix H (α,m) = (H (α,m) j,k ) ∈ C n×n with entries otherwise. (2.9) With the insights from the numerical methods for the Neumann eigenvalue problem on the unit interval, we have the following two sets of analytical eigenpairs.

Corner-overlapped block-diagonal matrices
In this section, we consider the following type of matrix, that is, We assume that the dimension of the matrix is ( n+ )×( n+ ). It is a block-diagonal matrix where the corners of blocks are overlapped. Therefore, we refer to this type of matrices as the corner-overlapped block-diagonal matrices.
In this section, we derive their analytical eigenpairs. To illustrate the idea, we consider the following matrix Alternatively, to nd its eigenvalues, we note that the rst and the last rows of (2.2) lead to On one hand, we solve these equations to arrive at which is then substituted into the third equation in (2.2) to get On the other hand, we substitute (2.20) and the third equation of (2.2) into the second and the fourth equations We now see that the EVP (2.2) is decomposed into two subproblems; that is, one EVP and QEVP as follows where I is the identity matrix (we assume that the dimension of I is adaptive to its occurrence, in this case, it is × ), Both matrices B and C are symmetric. The EVP (2.23) has an analytical eigenvalue λ = α which is one of the eigenvalues in (2.19). The characteristic polynomial of the QEVP (2.24) is which is a polynomial of order four. From fundamental theorem of algebra, it has four roots. It is easy to verify that λ j , j = , , , given in (2.19) are the four roots of the equation χ(x) = . Now, we propose the following idea. Assuming that λ ≠ , we rewrite the QEVP (2.24) as where which is a symmetric matrix. For a xed λ the matrix Ξ can be viewed as a constant matrix. We apply Theorem 2.1 with bandwidth m = and dimension n = . Thus, the eigenvalues of Ξ satisfy which is rewritten as a quadratic form in terms of λ j λ j − α + α + α cos(jπ/ ) λ j − α − α α + (α − α α ) cos(jπ/ ) = , j = , . (2.29) For j = , cos(jπ/ ) = / and we obtain the eigenvalues λ , as in (2.19) while for j = , cos(jπ/ ) = − / and we obtain the eigenvalues λ , as in (2.19). If λ = , then (2.27) is invalid and we note that a shift (divide (2.26) by λ − α for some non-zero α) will lead to the same set of eigenvalues. We now generalize the matrix A de ned (2.18) and consider the GEVP. Based on the idea above, we have the following theorem which gives analytical solutions to a class of GEVPs with corner-overlapped blockdiagonal matrices. Theorem 2.6 (Analytical eigenvalues and eigenvectors, set 5). Let A = G (α) and B = G (β) . Assume that B is invertible. Then, the GEVP (2.3) has n + eigenvalues (2.31) The corresponding eigenvectors are x j = (x j, , · · · , x j, n+ ) T with x j, k = c sin( j πkh), h = n + , j = , · · · , n, k = , , · · · , n, where · is the ceiling function.
The odd entries of the eigenvectors are given by (2.34) as This completes the proof.
Generalization of Theorem 2.6 is possible. The matrix de ned in (2.17) is a corner-overlapped blockdiagonal matrix where each block (except the rst and last) is of dimension × . One can generalize the block to be of dimension k × k where k ≥ . We give the following example where the block is × . The cubic FEM of the Laplacian eigenvalue problem with n uniform elements on the unit interval [ , ] leads to the sti ness and mass matrices [17,26] which are of dimension ( n − ) × ( n − ). To nd the analytical eigenvalues, we follow the same procedure as for quadratic FEM. By static condensation, the matrix eigenvalue problem GEVP Ku = λMu is rst rewritten to a block-matrix form A tt A ot Aoo where u t = (u , u , · · · , u n− ) T , u o = (u , u , u , u , · · · , u n− , u n− ) T . Similarly, det(Aoo) = leads to the eigenvalue λn = n , λ n = n . To nd the other eigenvalues, we rewrite A tt u t = as a cubic EVP which is of dimension (n − ) × (n − ). Herein, Λ = λh . Using the derivations from the proof of Theorem 2.6, we obtain the eigenvalues that are the roots of the equation where ζ = cos(jπh), j = , · · · , n − . The eigenvectors can be obtained similarly.

. Polynomial eigenvalue problems (PEVPs)
The PEVP is to nd the eigenpairs (λ ∈ C, x ∈ C n ) such that P(λ)x = with P(λ) := λ q Aq + λ q− A q− + · · · + A , (2.48) where A j ∈ C n×n , j = , , · · · , q. When q = and q = , the PEVP reduces to linear eigenvalue problem (LEVP) and QEVP, respectively. The LEVP is also referred to as the GEVP as discussed above. We note that the nonlinear eigenvalue problem Aee xe = de ned in the proof of Theorem 2.6 is a QEVP. In particular, the problem can be written as follows Herein, T (γ, ) is de ned as (2.1). The analytical eigenpairs are then derived based on the Theorem 2.1 with m = . The analytical eigenvalues of the cubic EVP (2.46) are derived in a similar fashion. We generalize these results and give the following result. We note that the above form is independent of the row number i and the eigenvector index number j. This means that a root of (2.51) and a vector x j = (x j, , · · · , x j,n ) T with x j,k = c sin(jπkh), k = , · · · , n, always satisfy the PEVP. Hence, the root and x j give an eigenpair. This completes the proof.
We remark that similar analytical results can be obtained if the Hankel matrix H is de ned as (

. Generalizations
We now consider some other generalizations. The simplest case is the constant scaling. Let c ≠ , c ≠ be constants, then the GEVP c Ax = λc Bx has eigenvalues λ =λc /c whereλ is an eigenvalue of the GEVP Ax =λBx. The eigenvectors remain the same. This constant scaling has applications in various numerical spectral approximations. For example, for FEM, the scaling is in the form ( /h)Ax = λhBx while for FDM, the scaling is c = /h , c = where h is the size of a mesh (cf., [25,26]). Following the book of Strang [25,Section 5], one may generalize these results to powers and products of matrices. For EVP of the form (2.2), the powers A k , k ∈ Z has eigenvalues λ k j , where λ j , j = , · · · , n are eigenvalues of (2.2). The eigenvectors remain the same. For the EVPs Ax = λx and Bx = µx, if A commutes with B, that is, AB = BA, then the two EVPs have the same eigenvectors and the eigenvalues of AB (or BA) are λµ. Additionally, in this case, A + B has eigenvalues λ + µ. For GEVP of the form (2.3), similar results can be obtained. If the matrices entries are commutative, then B − A = AB − . The GEVP A k x = µB k x has eigenvalues µ = λ k where λ is an eigenvalue of Ax = λBx. Similar results can be obtained for products and additions.

Remark 2.9.
Once the two sets of the eigenpairs are found, either numerically or analytically, the eigenpairs for the GEVP in the form (2.52) can be derived. A FEM (FDM, SEM, or IGA) discretization of −∆u = λu on unit square domain with a uniform tensor-product mesh leads to the GEVP in the form of (2.52). For three-or higher-dimensional problems, this result can be generalized in a similar fashion.

Trigonometric identities
In this section, we derive some trigonometric identities based on the eigenvector-eigenvalue identity that was rediscovered and coined recently in [13]. The eigenvector-eigenvalue identity for the EVP (2.2) is (see [13,Theorem 1]) where A is a Hermitian matrix with dimension n, (λ j , x j ), j = , · · · , n, are eigenpairs of Ax = λx with normalized eigenvectors x j = (x j, , · · · , x j,n ) T , and µ (k) l is an eigenvalue of A (k) y = µ (k) y with A (k) being the minor of A formed by removing the k th row and column. We generalize this identity for the GEVPs as follows.
Theorem 3.1 (Eigenvector-eigenvalue identity for the GEVP). Let A and B be Hermitian matrices with dimension n × n. Assume that B is invertible. Let (λ j , x j ), j = , · · · , n, be the eigenpairs of the GEVP (2.3) with normalized eigenvectors x j = (x j, , · · · , x j,n ) T . Then, there holds

2)
where µ (k) l is an eigenvalue of A (k) y = µ (k) B (k) y with A (k) and B (k) being minors of A and B formed by removing the k th row and column, respectively, η l is an eigenvalue of By = ηy, and η (k) l is an eigenvalue of B (k) y = η (k) y.
Proof. We follow the proof of (3.1) for Ax = λx that uses perturbative analysis in [13,Sect. 2.4] ( rst appeared in [21]). Firstly, since B is invertible, det(B) ≠ and hence η l ≠ , l = , · · · , n. Let Q(λ) be the characteristic polynomial of the GEVP (2.3). Then, Similarly, let P (k) (µ (k) ) be the characteristic polynomial of the GEVP A (k) y = µ (k) B (k) y. Then, Now, with the limiting argument, we assume that A has simple eigenvalues. Let ϵ be a small parameter and we de ne the perturbed matrix A ϵ,k = A + ϵe k e T k , k = , · · · , n, (3.6) where {e k } n k= is the standard basis. The perturbed GEVP is de ned as Using (3.3) and cofactor expansion, the characteristic polynomial of this perturbed GEVP can be expanded as With x j being a normalized eigenvector, one has x T j · x j = , x T j Bx j = η j , j = , · · · , n. (3.9) Using this normalization, from perturbation theory, the eigenvalue λ ϵ j of (3.7) can be expanded as Applying the Taylor expansion and Q(λ j ) = , we rewrite which the linear term in ϵ leads to |x j,k | Q (λ j ) = η j P (k) (λ j ). We note that this identity is independent of the matrix entries α and α . The left hand side can be written in terms of a cosine function as n+ ( − cos kπ n+ ) to have an identity in terms of only cosine functions. For example, let n = , k = , then the identity boils down to = sin π = cos(π/ ) − cos(π/ ) cos(π/ ) − cos( π/ ) = / − / + / = . (3.14) Similarly, we have for n ≥ , k = , · · · , n, l = , · · · , n − n + sin klπ n + = It is obvious that (3.16) reduces to (3.15) when B is an identity matrix (or multiplied by a nonzero constant).

Remark 3.2.
Other similar trigonometric identities can be established. Moreover, Theorems 2.1-2.6 give various analytical eigenpairs. An application of the eigenvector-eigenvalue identity (3.1) along with these analytical results sets up a system of equations governing the eigenvalues of the minors of the original matrices. Thus, the eigenvalues of these minors can be found by solving the system of equations.

Concluding remarks
We rst remark that the ideas of nding analytical solutions to the GEVPs with Toeplitz-plus-Hankel and corner-overlapped block-diagonal matrices can be applied to solve other problems where a particular solution form is sought. Other applications include matrix representations of di erential operators such as the Schrödinger operator in quantum mechanics [11] and the n-order operators [12]. Moreover, the idea to solve QEVP can be applied to solve other nonlinear EVPs. The boundary modi cations in the Toeplitz-plus-Hankel matrices give new insights for designing better numerical methods. For example, the high-order IGA (cf., [16]) produces outliers in the high-frequency region of the spectrum. A method which modi es the boundary terms to arrive at the Toeplitz-plus-Hankel matrices will be outlier-free [9]. For FDM, the structure of the Toeplitz-plus-Hankel matrices give insights to the design better higher-order approximations near the domain boundaries. Lastly, we remark that the corner-overlapped block-diagonal matrices have applications in the FEMs and the discontinuous Galerkin methods.