The generalized inverse eigenvalue problem of Hamiltonian matrices and its approximation

1.   Introduction

Throughout this paper, $\mathbb{C}^{m\times n}$, $\mathbb{R}^{m\times n}$, $\mathbb{OR}^{n\times n}$ and $\mathbb{SR}^{n\times n}$ stand for the sets of all $m \times n$ complex matrices, all $m \times n$ real matrices, all $n \times n$ orthogonal matrices and all $n \times n$ real-valued symmetric matrices, respectively. The symbol $A^{\top}$ and tr$(A)$ stand for the transpose and the trace of a matrix $A$, respectively. $I_n$ represents the identity matrix of size $n$, and $\mathbb{HR}^{2n\times 2n}$ represents the set of all $2n \times 2n$ Hamiltonian matrices, that is, $\mathbb{HR}^{2n\times 2n} = \left\{A\left|(AJ)^{\top} = AJ, \ A\in \mathbb{R}^{2n\times 2n} \right.\right\},$ where $J = \left[ \begin{array}{cc} 0 & I_n \\ -I_n & 0 \\ \end{array} \right] \in \mathbb{R}^{2n\times 2n}$.

Hamiltonian matrices are widely applied in Hamiltonian systems of differential equations ^[1,2], optimal quadratic linear control ^[3] and $H_{\infty}$ optimization ^[4], etc. For example, Hamiltonian matrix elements from a symmetric wave function are necessary to study the structure of deuterated molecules ^[5]. Also, the eigenvalue problems for Hamiltonian and skew-Hamiltonian matrices appear frequently in scientific and engineering applications. Such as to compute the conformal parameterization via a constrained energy minimization problem in the field of digital geometry processing ^[6,7], quantum mechanical problems with time reversal symmetry ^[8,9], the study of closed shell Hartree-Fock wave functions in response theory ^[10,11] and total least squares problems with symmetric constraints ^[12].

Inverse eigenvalue problems emerge from many application areas, and have been studied by many scholars ^{[13,14,15,16,17,18]}. Generalized inverse eigenvalue problems are concerned in structural dynamics ^[19,20,21], parameter identification ^[22] and pole assignment ^[23], etc. Recently, Zhao and Zhang ^[24] derived the solvability conditions for the inverse eigenvalue problem of normal skew $J$-Hamiltonian matrices by the Moore-Penrose generalized inverse and the generalized singular value decomposition. Zhang and Yuan ^[25] solved the generalized inverse eigenvalue problems of Hermitian and $J$-Hamiltonian/skew-Hamiltonian matrices by applying the singular value decomposition and the spectral decomposition. However, the problem of OAP cannot be considered due to the complexity of the expression of the general solution. Very recently, Yuan and Chen ^[26] solved the inverse eigenvalue problem and the optimal approximation problem for Hamiltonian matrices by using the generalized singular value decomposition. Nevertheless, the generalized inverse eigenvalue problem of Hamiltonian matrices seems rarely to be discussed in the literatures, which motivates us to study such kind of inverse problem and the associated approximation problem. That is, in this paper, we will consider the following generalized inverse eigenvalue problem and the associated optimal approximation problem, which is a generalization of the problems discussed in ^[26].

Problem GIEP. Given a pair of matrices $(\Lambda, X)$ in the form $\Lambda = {\rm{diag}}\{\lambda_1, \cdots, \lambda_{p}\}\in \mathbb{C}^{p\times p},$ and $X = [{\bf{x}}_{1}, \cdots, {\bf{x}}_{p}]\in \mathbb{C}^{2n\times p}$, where diagonal elements of $\Lambda$ are all distinct, $X$ is of full column rank $p$, and both $\Lambda$ and $X$ are closed under complex conjugation in the sense that $\lambda_{2i} = \bar{\lambda}_{2i-1}\in \mathbb{C},$ ${\bf{x}}_{2i} = \bar{{\bf{x}}}_{2i-1}\in \mathbb{C}^{2n}$ for $i = 1, \cdots, l,$ and $\lambda_{j}\in \mathbb{R},$ ${\bf{x}}_{j}\in \mathbb{R}^{2n}$ for $j = 2l+1, \cdots, p.$ Find $A, B\in \mathbb{HR}^{2n\times 2n}$ such that

Problem OAP. Given $\tilde{A}, \tilde{B}\in \mathbb{R}^{2n\times 2n}$. Find $(\hat{A}, \hat{B}) \in \mathbb{S_{E}}$ such that

where ${\|\cdot \|}$ is the Frobenius norm and $\mathbb{S_{E}}$ is the solution set of Problem GIEP.

By using the QR-decomposition, the representation of the general solution of Problem GIEP is deduced and the unique optimal approximation solution $(\hat{A}, \hat{B})$ of Problem OAP is obtained. Finally, two numerical examples are presented to illustrate the efficiency of the results.

2.   The solution of Problem GIEP

Define $T_{p}$ as

where $i = \sqrt{-1}.$ It is easy to verify that $T_{p}$ is a unitary matrix, that is, $\bar{T}_{p}^{\top}T_{p} = I_{p}.$ With this matrix, we have

where $\alpha_{i}$ and $\beta_{i}$ are the real part and imaginary part of the complex number $\lambda_{i}$, and ${\bf{y}}_{i}$ and ${\bf{z}}_{i}$ are, respectively, the real part and imaginary part of the complex vector ${\bf{x}}_{i}$ for $i = 1, 3, \cdots, 2l-1.$ Then, Eq $(1.1)$ can be equivalently written as

clearly, Eq $(2.3)$ is equivalent to

Since rank$(X)$ = rank$(\tilde{X})$ = $p$, the QR-decomposition of $J^{\top}\tilde{X}$ is of the form

where $Q = [Q_{1}, Q_{2}]\in \mathbb{OR}^{2n\times 2n}$ with $Q_{1}\in \mathbb{R}^{2n\times p}$, and $R\in \mathbb{R}^{p\times p}$ is nonsingular. Partition the parameter matrices $Q^{\top}AJQ$ and $Q^{\top}BJQ$ into blocks:

where $A_{11}, \ A_{22}, \ B_{11}$ and $B_{22}$ are real-valued symmetric matrices. By $(2.5)$ and $(2.6)$, Eq $(2.4)$ is equivalent to

Then, it follows from Eq $(2.7)$ that

By Eq $(2.8)$, $B_{11}$ is a symmetric matrix implies that

Write

then Eq $(2.10)$ can be written as

By direct calculation, we have

where $a_{2i-1},$ $b_{2i-1},$ $i = 1, \cdots, l,$ and $c_{j},$ $j = 2l+1, \cdots, p,$ are arbitrary real numbers. Thus

Combining $(2.8)$ with $(2.11)$, we find that

By Eq $(2.9),$ we have

where $A_{12}\in \mathbb{R}^{p\times (2n-p)}$ is an arbitrary matrix.

Summing up above discussion, we can obtain the following result.

Theorem 2.1. Suppose that $\Lambda = \mathit{{\rm{diag}}}\{\lambda_1, \cdots, \lambda_{p}\}\in \mathbb{C}^{p\times p},$ $X = [{\bf{x}}_{1}, \cdots, {\bf{x}}_{p}]\in \mathbb{C}^{2n\times p}$, where diagonal elements of $\Lambda$ are all distinct, $X$ is of full column rank $p$, and both $\Lambda$ and $X$ are closed under complex conjugation. Let the real matrices $\tilde{\Lambda}$ and $\tilde{X}$ be given by $(2.1)$ and $(2.2)$ and the QR-decomposition of $J^{\top}\tilde{X}$ be given by $(2.5)$. Then the general solution of Problem GIEP can be expressed as

where $A_{12}\in \mathbb{R}^{p\times (2n-p)}$, $A_{22}\in \mathbb{SR}^{(2n-p)\times (2n-p)}$ and $B_{22}\in \mathbb{SR}^{(2n-p)\times (2n-p)}$ are arbitrary matrices, and $C$ is given by $(2.13)$.

3.   The solution of Problem OAP

According to $(2.17),$ we know that the solution set $\mathbb{S_E}$ is always nonempty and $\mathbb{S_E}$ is a closed convex subset, which implies that Problem OAP has a unique solution $(\hat{A}, \hat{B})\in \mathbb{S_E}$ by the optimal approximation theorem (see Ref. ^[27]). For the given matrices $\tilde{A}, \tilde{B}\in \mathbb{R}^{2n\times 2n},$ write

then

Therefore, $\|A-\tilde{A}\|^2+\|B -\tilde{B}\|^2$ = min if and only if

Let

where

and $R_{1, 2i-1}\in \mathbb{R}^{2\times p},$ $R_{2, j}\in \mathbb{R}^{1\times p} \ (i = 1, \cdots, l, j = 2l+1, \cdots, p).$ Furthermore, let

where

Then the relation of $(3.2)$ is equivalent to

that is,

where $g_{m, n} = \rm{tr}(D_{m}^{\top}D_{n})+\rm{tr}(E_{m}^{\top}E_{n}),$ $h_{m} = \rm{tr}(D_{m}^{\top}\tilde{A}_{11})+\rm{tr}(E_{m}^{\top}\tilde{B}_{11}),$ $e = \rm{tr}(\tilde{A}_{11}^{\top}\tilde{A}_{11})+\rm{tr}(\tilde{B}_{11}^{\top}\tilde{B}_{11}),$ $m, n = 1, \cdots, p$.

Consequently,

Clearly, $f(a_{1}, b_{1}, \cdots, a_{2l-1}, b_{2l-1}, c_{2l+1}, \cdots, c_{p})$ = min if and only if

Therefore,

If let

where $G$ is symmetric matrix. Then Eq $(3.8)$ is equivalent to

and the solution of Eq $(3.9)$ is

Substituting $(3.10)$ into $(2.13)$, we can obtain $C$ explicitly. Similarly, Eq $(3.3)$ is equivalent to

Thus,

setting $\frac{\partial f(A_{12})}{\partial A_{12}} = 0,$ we obtain

where $P = R\tilde{\Lambda}R^{-1}.$ $A_{22},$ $B_{22}$ are symmetric matrices implies that the relations of $(3.4)$ and $(3.5)$ are equivalent to

therefore, we have

Theorem 3.1. Given $\tilde{A}, \tilde{B}\in \mathbb{R}^{2n\times 2n},$ then the Problem OAP has a unique solution and the unique solution of Problem OAP is

where $A_{12}$ and $\ A_{22}, \ B_{22}$ are given by $(3.11)$ and $(3.12),$ and $a_{1}, b_{1}, \cdots, a_{2l-1}, b_{2l-1}, c_{2l+1}, \cdots, c_{p}$ are given by $(3.10)$, respectively.

4.   Numerical examples

According to Theorems 2.1 and 3.1, we have the following algorithm for solving Problem OAP.

Algorithm 4.1.

1). Input $\Lambda, \ X, \ J, \ \tilde{A}, \ \tilde{B}.$

2). Compute real-valued matrices $\tilde{\Lambda}, \ \tilde{X}$ by $(2.1)$ and $(2.2)$, respectively.

3). Compute the QR-decomposition of the matrix $J^{\top}\tilde{X}$ by $(2.5)$.

4). Compute $\tilde{A}_{ij}$, $\tilde{B}_{ij}$ by $(3.1),$ $i, j = 1, 2$.

5). Compute $R^{-1}$ by $(3.6)$ to form $R_{1}, R_{2}$.

6). Compute $D_{2i-1}, D_{2i}, D_{j}$, $E_{2i-1}, E_{2i}$ and $E_{j}\ (i = 1, \cdots, l, j = 2l+1, \cdots, p$) by $(3.7).$

7). Compute $g_{m, n} = \mbox{tr}(D_{m}^{\top}D_{n})+\mbox{tr}(E_{m}^{\top}E_{n})$ and $h_{m} = \mbox{tr}(D_{m}^{\top}\tilde{A}_{11})+\mbox{tr}(E_{m}^{\top}\tilde{B}_{11})$ ($m, n = 1, \cdots, p$).

8). Compute $a_{1}, b_{1}, \cdots, a_{2l-1}, b_{2l-1}, c_{2l+1}, \cdots, c_{p}$ by $(3.10)$.

9). Compute $A_{12}$ by $(3.11).$

10). Compute $A_{22}$ and $B_{22}$ by $(3.12)$.

11). Compute $\hat{A}$ and $\hat{B}$ by $(3.13).$

Remark 4.1. After statistics, we find that the amount of computations required by Algorithm 1 is about $p^5+p^4+\frac{19}{3}p^3+14np^2+64n^3$ flops.

Example 4.1. Let $n = 5, \ p = 5$, and the matrices $\Lambda, \ X, \ \tilde{A}$ and $\tilde{B}$ be given by

By applying Algorithm $4.1$, we can obtain the unique solution $(\hat{A}, \hat{B})$ of Problem OAP as follows:

and

which implies that $\hat{A} X \Lambda = \hat{B}X$ reproduces the desired eigenvalues and eigenvectors.

Example 4.2. We consider an inverse problem for the spectral conformal parameterization (see Refs.^[6,7]). Let $n = 5, \ p = 4$, and the matrices $\Lambda, \ X, \ \tilde{B}$ and $\tilde{L}_{C}$ be given by

By calculating, we can obtain the unique solution $(\hat{B}, \hat{L}_{C})$ of Problem OAP as follows:

Furthermore, we can obtain the following numerical results: Therefore, the new model $\hat{B} X \Lambda = \hat{L}_{C}X$ reproduces the desired eigenvalues and eigenvectors.

Acknowledgments

The authors would like to express their gratitude to the anonymous referees for their valuable suggestions and comments for the revision of this manuscript.

Conflict of interest

The authors declare no conflict of interest.