1 Introduction

In this paper, we mainly discuss the following partially described inverse eigenvalue problem which is considered in linear manifold.

Problem 1

Given the partial eigeninformation \(Y = (y_{1},y_{2},\ldots,y_{m}) \in {\mathbb{C}}^{n\times m}\) and \(\Lambda= \operatorname{diag}(\lambda_{1},\lambda_{2}, \ldots,\lambda_{m}) \in {\mathbb{C}}^{m\times m}\), consider the set

$$\mathcal{M}(Y,\Lambda):= \{A\in\Omega| AY = Y\Lambda\} $$

of matrices A maintaining the eigeninformation, where Ω is the set of certain n-by-n structured matrices.

The above problem usually appears in the design and modification of mass-spring systems, dynamic structures, Hopfield neural networks, vibration in mechanic, civil engineering, and aviation [2–4]. Furthermore, the inverse eigenvalue problems involving Hamiltonian matrices have drawn considerable interest. For example, Zhang et al. [5] solved the inverse eigenvalue problem of Hermitian and generalized Hamiltonian matrices. Then Bai [6] settled the case of Hermitian and generalized skew-Hamiltonian matrices. Xie et al. [7] resolved the case of symmetric skew-Hamiltonian matrices. Qian and Tan [8] also considered the cases of Hermitian and generalized Hamiltonian/skew-Hamiltonian matrices from different perspectives. But the Hamiltonian matrices they considered are the special cases of the following normal J-Hamiltonian matrices and normal skew J-Hamiltonian matrices.

In the following, let \(I_{n}\) be the \(n\times n\) identity matrix.

Definition 1

([1])

Given a normal matrix \(J \in{\mathbb {R}}^{n\times n}\) with \(J^{2}=-I_{n}\). A matrix \(A \in{\mathbb {C}}^{n\times n}\) is referred to as normal J-Hamiltonian if and only if \({(AJ)}^{*}=AJ\) and \(AA^{*}=A^{*}A\).

Definition 2

Let \(J \in{\mathbb{R}}^{n\times n}\) be a normal matrix such that \(J^{2}=-I_{n}\). A matrix \(A \in{\mathbb{C}}^{n\times n}\) is called normal skew J-Hamiltonian if and only if \({(AJ)}^{*}=-AJ\) and \(AA^{*}=A^{*}A\). The set of all \(n\times n\) normal skew J-Hamiltonian matrices is denoted by \({\mathcal{NS}}^{n\times n}(J)\).

In the above definitions, \(A^{*}\) signifies the conjugate transpose of a matrix \(A \in{\mathbb{C}}^{n\times n}\). It is obvious that J is a real orthogonal skew-symmetric matrix, i.e., \(J=-J^{T}=-J^{-1}\). This indicates that \(n=2k\), \(k\in\mathbb{N}\). The above Hamiltonian matrices are also of importance in several engineering areas such as optimal quadratic linear control, \(H_{\infty}\) optimization, and the related problem of solving Riccati algebraic equations [9].

Recently, Gigola et al. [1] solved Problem 1 for normal J-Hamiltonian matrices. In this paper, we present a set of alternative conditions assuring the solvability of the problem that involves skew normal J-Hamiltonian matrices. In order to present more simple conditions to be verified, we mainly use Sun’s [10] and Penrose’s [11] results and the generalized singular value decomposition to solve Problem 1 when the set \(\Omega={\mathcal{NS}}^{n\times n}(J)\). A similar technique may be used to solve the inverse eigenvalue problem for normal J-Hamiltonian matrices.

2 Preliminaries

Throughout this paper, we denote by \(\operatorname{rank}(A)\) and \(A^{\dagger}\) the rank and Moore–Penrose generalized inverse of a matrix \(A \in{\mathbb {C}}^{n\times m}\), respectively. \(I_{n}\), 0, and \(\mathrm {i}=\sqrt{-1}\) respectively signify the identity matrix of order n, a zero matrix or a vector with appropriate size, and the imaginary unit. Moreover, for any matrix \(A \in{\mathbb{C}}^{n\times m}\), \({\mathscr {R}}_{A}=I_{n}-AA^{\dagger}\) and \({\mathscr{L}}_{A}=I_{m}-A^{\dagger}A\) signify specified orthogonal projectors.

Firstly, we consider the structure of the normal skew J-Hamiltonian matrices.

Lemma 1

Let \(J \in{\mathbb{R}}^{n\times n}\) be a normal matrix such that \(J^{2}=-I_{n}\). Then a matrix \(A \in{\mathbb{C}}^{n\times n}\) is normal skew J-Hamiltonian if and only if

$$ A=U\left [ \textstyle\begin{array}{c@{\quad}c} A_{11} & A_{12}\\ -A^{*}_{12} & A_{22} \end{array}\displaystyle \right ]U^{*}, $$
(2.1)

where \(n=2k\), \(k\in\mathbb{N}\), \(A_{11}=A^{*}_{11}\in{\mathbb{C}}^{k\times k}\), \(A_{22}=A^{*}_{22}\in{\mathbb{C}}^{k\times k}\), \(A_{11}A_{12}=A_{12}A_{22}\), and \(U \in{\mathbb{C}}^{n\times n}\) is a unitary matrix such that

$$ U^{*}JU=\left [ \textstyle\begin{array}{c@{\quad}c} \mathrm{i}I_{k} & \bf{0}\\ \bf{0} & -\mathrm{i}I_{k} \end{array}\displaystyle \right ]. $$
(2.2)

Proof

Because \(J \in{\mathbb{R}}^{n\times n}\) is a normal matrix and \(J^{2}=-I_{n}\), then J is a real orthogonal skew-symmetric matrix. Therefore, there exists a unitary matrix \(U \in{\mathbb{C}}^{n\times n}\) such that (2.2) holds, where \(n=2k\), \(k\in\mathbb{N}\).

Then partition \(U^{*}AU\) conforms with (2.2) as

$$ U^{*}AU=\left [ \textstyle\begin{array}{c@{\quad}c} A_{11} & A_{12}\\ A_{21} & A_{22} \end{array}\displaystyle \right ]. $$
(2.3)

From Definition 2, we know that \(JAJ=-A^{*}\). It follows that

$$\left [ \textstyle\begin{array}{c@{\quad}c} -A_{11} & A_{12}\\ A_{21} & -A_{22} \end{array}\displaystyle \right ]=-\left [ \textstyle\begin{array}{c@{\quad}c} A^{*}_{11} & A^{*}_{21}\\ A^{*}_{12} & A^{*}_{22} \end{array}\displaystyle \right ]. $$

Thus we have

$$ A_{11}=A^{*}_{11},\qquad A_{22}=A^{*}_{22},\qquad A_{12}=-A^{*}_{21}. $$
(2.4)

Because \(AA^{*}=A^{*}A\), then from (2.3) and (2.4) we have \(A_{11}A_{12}=A_{12}A_{22}\). Therefore, (2.1) holds. □

Then we introduce the following results to solve Problem 1 later on.

Lemma 2

(Sun [10])

Let \(A_{1},B_{1}\in{\mathbb {C}}^{n\times m}\) be given. The linear matrix equation \(X_{1}A_{1}=B_{1}\) has a Hermitian solution \(X_{1}\in{\mathbb{C}}^{n\times n}\) if and only if

$$B_{1}{\mathscr{L}}_{A_{1}}=0,\qquad A^{*}_{1}B_{1}=B^{*}_{1}A_{1}. $$

In this case, the general solution can be expressed as

$$X_{1}=B_{1}A_{1}^{\dagger}+ { \bigl(B_{1}A_{1}^{\dagger}\bigr)}^{*} - \frac{1}{2}{\bigl(A_{1}^{\dagger}\bigr)}^{*} \bigl(A^{*}_{1}B_{1} + B^{*}_{1}A_{1} \bigr)A_{1}^{\dagger}+ {\mathscr{R}}_{A_{1}}R_{1}{ \mathscr {R}}_{A_{1}}, $$

where \(R_{1}\in{\mathbb{C}}^{n\times n}\) is an arbitrary Hermitian matrix.

In this lemma, the general solution can be also expressed as

$$X_{1}=B_{1}A_{1}^{\dagger}+ { \bigl(B_{1}A_{1}^{\dagger}\bigr)}^{*}{ \mathscr{R}}_{A_{1}}+{\mathscr {R}}_{A_{1}}R_{1}{ \mathscr{R}}_{A_{1}}. $$

The following lemma is taken from [11], see Corollary 2 in [11].

Lemma 3

(Penrose [11])

Let \(A_{2}\in{\mathbb {C}}^{n\times m}\), \(C_{2}\in{\mathbb{C}}^{n\times p}\), \(B_{2}\in{\mathbb {C}}^{p\times q}\), and \(D_{2}\in{\mathbb{C}}^{m\times q}\) be given. The pair of matrix equations \(A_{2}X_{2}=C_{2}\), \(X_{2}B_{2}=D_{2}\) has a solution \(X_{2}\in{\mathbb{C}}^{m\times p}\) if and only if

$${\mathscr{R}}_{A_{2}}C_{2}=0,\qquad D_{2}{ \mathscr{L}}_{B_{2}}=0,\qquad A_{2}D_{2}=C_{2}B_{2}. $$

Moreover, the general solution can be expressed as

$$X_{2}=A_{2}^{\dagger}C_{2} + { \mathscr{L}}_{A_{2}}D_{2}B_{2}^{\dagger}+ { \mathscr {L}}_{A_{2}}R_{2}{\mathscr{R}}_{B_{2}}, $$

where \(R_{2}\in{\mathbb{C}}^{m\times p}\) is an arbitrary matrix.

3 Solvability conditions and general solution of Problem 1

Given a normal matrix \(J \in{\mathbb{R}}^{n\times n}\) with \(J^{2}=-I_{n}\), let \(Y \in{\mathbb{C}}^{n\times m}\) and \(\Lambda\in {\mathbb{C}}^{m\times m}\) be given in Problem 1. In order to solve this problem for the case of normal skew J-Hamiltonian matrices, we need to obtain the normal skew J-Hamiltonian solution of the linear matrix equation

$$ AY=Y\Lambda. $$
(3.1)

If equation (3.1) is consistent, then the set \(\mathcal {M}(Y,\Lambda)\) is nonempty. By Lemma 1, equation (3.1) is equivalent to the following:

$$ \left [ \textstyle\begin{array}{c@{\quad}c} A_{11} & A_{12}\\ -A^{*}_{12} & A_{22} \end{array}\displaystyle \right ]U^{*}Y=U^{*}Y \Lambda. $$
(3.2)

Let

$$ U^{*}Y=\left [ \textstyle\begin{array}{c} Y_{1} \\ Y_{2} \end{array}\displaystyle \right ],\quad Y_{1} \in{\mathbb{C}}^{k\times m}, Y_{2} \in{\mathbb {C}}^{k\times m}. $$
(3.3)

Then (3.2) can be rewritten as follows:

$$ \left\{ \textstyle\begin{array}{l}A_{11}Y_{1}+A_{12}Y_{2} = Y_{1}\Lambda,\\ -A^{*}_{12}Y_{1}+A_{22}Y_{2} = Y_{2}\Lambda. \end{array}\displaystyle \right.$$
(3.4)

Thus we have

$$ \left\{ \textstyle\begin{array}{l} A_{12}Y_{2} = Y_{1}\Lambda-A_{11}Y_{1},\\ Y^{*}_{1}A_{12} = Y^{*}_{2}A_{22}-\Lambda^{*}Y^{*}_{2}. \end{array}\displaystyle \right. $$

By Lemma 3, the above system of matrix equations has a solution \(A_{12} \in{\mathbb{C}}^{k\times k}\) if and only if

$$ \left\{ \textstyle\begin{array}{l} A_{11}Y_{1}{\mathscr{L}}_{Y_{2}}=Y_{1}\Lambda{\mathscr{L}}_{Y_{2}},\\ A_{22}Y_{2}{\mathscr{L}}_{Y_{1}}=Y_{2}\Lambda{\mathscr{L}}_{Y_{1}}, \quad\mbox{where } {\mathscr{L}}_{Y_{1}}=I_{k}-Y_{1}Y_{1}^{*},\\ Y^{*}_{1}A_{11}Y_{1}+Y^{*}_{2}A_{22}Y_{2}=Y^{*}_{1}Y_{1}\Lambda+\Lambda^{*}Y^{*}_{2}Y_{2}. \end{array}\displaystyle \right. $$
(3.5)

Then, by Lemma 2, the first equation in (3.5) has a Hermitian solution \(A_{11}\in{\mathbb{C}}^{k\times k}\) if and only if

$$ Y_{1}\Lambda{\mathscr{L}}_{Y_{2}}{ \mathscr{L}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}=0,\qquad {\mathscr{L}}_{Y_{2}}\bigl(Y^{*}_{1}Y_{1} \Lambda-\Lambda^{*}Y^{*}_{1}Y_{1}\bigr){\mathscr{L}}_{Y_{2}}=0. $$
(3.6)

In this case, the general solution is

$$ A_{11}=Y_{1}\Lambda{\mathscr{L}}_{Y_{2}}{(Y_{1}{ \mathscr{L}}_{Y_{2}})}^{\dagger}+ {\bigl({\mathscr{L}}_{Y_{2}}Y^{*}_{1} \bigr)}^{\dagger}{\mathscr{L}}_{Y_{2}}\Lambda ^{*}Y^{*}_{1}{ \mathscr{R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}+ {\mathscr {R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}S_{1}{ \mathscr{R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}, $$
(3.7)

where \(S_{1}\in{\mathbb{C}}^{k\times k}\) is an arbitrary Hermitian matrix.

Similarly, by Lemma 2, the second equation in (3.5) has a Hermitian solution \(A_{22}\in{\mathbb{C}}^{k\times k}\) if and only if

$$ Y_{2}\Lambda{\mathscr{L}}_{Y_{1}}{ \mathscr{L}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}=0,\qquad {\mathscr{L}}_{Y_{1}}\bigl(Y^{*}_{2}Y_{2} \Lambda-\Lambda^{*}Y^{*}_{2}Y_{2}\bigr){\mathscr{L}}_{Y_{1}}=0. $$
(3.8)

In this case, the general solution is

$$ A_{22}=Y_{2}\Lambda{\mathscr{L}}_{Y_{1}}{(Y_{2}{ \mathscr{L}}_{Y_{1}})}^{\dagger}+ {\bigl({\mathscr{L}}_{Y_{1}}Y^{*}_{2} \bigr)}^{\dagger}{\mathscr{L}}_{Y_{1}}\Lambda ^{*}Y^{*}_{2}{ \mathscr{R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}+ {\mathscr {R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}S_{2}{ \mathscr{R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}, $$
(3.9)

where \(S_{2}\in{\mathbb{C}}^{k\times k}\) is an arbitrary Hermitian matrix. Let

$$ \begin{aligned}[b]G ={}& Y^{*}_{1}Y_{1}\Lambda+ \Lambda^{*}Y^{*}_{2}Y_{2}-Y^{*}_{1}\bigl[Y_{1} \Lambda{\mathscr {L}}_{Y_{2}}{(Y_{1}{\mathscr{L}}_{Y_{2}})}^{\dagger}+ {\bigl({\mathscr {L}}_{Y_{2}}Y^{*}_{1}\bigr)}^{\dagger}{ \mathscr{L}}_{Y_{2}}\Lambda^{*}Y^{*}_{1}{\mathscr {R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}\bigr]Y_{1} \\ & -Y^{*}_{2}\bigl[Y_{2}\Lambda{\mathscr{L}}_{Y_{1}}{(Y_{2}{ \mathscr{L}}_{Y_{1}})}^{\dagger}+ {\bigl({\mathscr{L}}_{Y_{1}}Y^{*}_{2} \bigr)}^{\dagger}{\mathscr{L}}_{Y_{1}}\Lambda ^{*}Y^{*}_{2}{ \mathscr{R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}\bigr]Y_{2}.\end{aligned} $$
(3.10)

From (3.5), (3.7), and (3.9) we know that \(G=G^{*}\) is equivalent to

$$Y^{*}_{1}Y_{1}\Lambda+\Lambda^{*}Y^{*}_{2}Y_{2}= \Lambda^{*}Y^{*}_{1}Y_{1}+Y^{*}_{2}Y_{2} \Lambda. $$

Then substituting (3.7) and (3.9) into the third equation in (3.5) yields

$$ Y^{*}_{1}{\mathscr{R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}S_{1}{ \mathscr {R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}Y_{1} +Y^{*}_{2}{ \mathscr{R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}S_{2}{\mathscr {R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}Y_{2}=G. $$
(3.11)

Thus we need to obtain a pair of Hermitian solutions \(( \widehat{S}_{1}, \widehat{S}_{2})\) of the linear matrix equation (3.11). Firstly, we give the generalized singular value decomposition (GSVD) of the matrix pair \(({\mathscr{R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}Y_{1}, {\mathscr {R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}Y_{2})\) as follows (see, for example, [12]):

$$ {\mathscr{R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}Y_{1}=U_{1} \Pi_{1}M, \qquad{\mathscr {R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}Y_{2}=U_{2} \Pi_{2}M, $$
(3.12)

where \(U_{1}\) and \(U_{2}\) are unitary matrices of order k and \(M \in {\mathbb{C}}^{m\times m}\) is a nonsingular matrix, and

$$\begin{gathered} \Pi_{1}= \textstyle\begin{array}{cl} \left ( \textstyle\begin{array}{c@{\qquad}c@{\qquad}c@{\qquad}c} I_{r_{3}-r_{2}} & \qquad\bf{0} & \qquad\bf{0} & \quad\bf{0}\\ \bf{0} & \qquad\Lambda_{1} & \qquad\bf{0} & \quad\bf{0}\\ \bf{0} & \qquad\bf{0} & \qquad\bf{0} & \quad\bf{0} \end{array}\displaystyle \right ) & \left . \textstyle\begin{array}{l} r_{3}-r_{2} \\ r_{1}+r_{2}-r_{3}\\ k-r_{1} \end{array}\displaystyle \right . \\ \left . \textstyle\begin{array}{@{\quad}c@{\qquad}c@{\quad}c@{\quad}c} r_{3}-r_{2} & r_{1}+r_{2}-r_{3} & r_{3}-r_{1} & m-r_{3} \end{array}\displaystyle \right . & \end{array}\displaystyle , \\\Pi_{2}= \textstyle\begin{array}{cl} \left ( \textstyle\begin{array}{c@{\qquad}c@{\qquad}c@{\qquad}c} \bf{0} & \quad\hspace{6pt}\bf{0} & \quad\bf{0} & \quad\bf{0}\\ \bf{0} & \quad\hspace{6pt}\Lambda_{2} & \quad\bf{0} & \quad\bf{0}\\ \bf{0} & \quad\hspace{6pt}\bf{0} & \quad I_{r_{3}-r_{1}} & \quad\bf{0} \end{array}\displaystyle \right ) & \left . \textstyle\begin{array}{l} k-r_{2} \\ r_{1}+r_{2}-r_{3}\\ r_{3}-r_{1} \end{array}\displaystyle \right . \\ \left . \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} r_{3}-r_{2} & r_{1}+r_{2}-r_{3} & r_{3}-r_{1} & \quad m-r_{3} \end{array}\displaystyle \right . & \end{array}\displaystyle \end{gathered} $$

are block matrices with the same column partitioning. In the matrices \(\Pi_{1}\) and \(\Pi_{2}\),

$$\begin{aligned}& r_{1} = \operatorname{rank}({\mathscr{R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}Y_{1}),\qquad r_{2} = \operatorname{rank}({\mathscr{R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}Y_{2}),\\& r_{3} = \operatorname{rank}\bigl(Y^{*}_{1}{\mathscr{R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}},Y^{*}_{2}{ \mathscr {R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}\bigr), \\& \Lambda_{1} = \operatorname{diag}(\xi_{1},\xi_{2}, \ldots,\xi_{r_{1}+r_{2}-r_{3}}),\quad 1>\xi _{1}\geq\cdots\geq \xi_{r_{1}+r_{2}-r_{3}}>0, \\& \Lambda_{2} = \operatorname{diag}(\eta_{1},\eta_{2}, \ldots,\eta_{r_{1}+r_{2}-r_{3}}),\quad 0< \eta_{1}\leq\cdots\leq \eta_{r_{1}+r_{2}-r_{3}}< 1, \\& \Lambda^{2}_{1}+\Lambda^{2}_{2}=I_{r_{1}+r_{2}-r_{3}}. \end{aligned}$$

We further partition the nonsingular matrix

$$M^{-1}=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} M_{1} & M_{2} & M_{3} & M_{4}\\ r_{3}-r_{2} & r_{1}+r_{2}-r_{3} & r_{3}-r_{1} & m-r_{3} \end{array}\displaystyle \right ) $$

compatibly with the block column partitioning of \(\Pi_{1}\) or \(\Pi_{2}\). Denote

$$ {\bigl(M^{*}\bigr)}^{-1}GM^{-1}=(G_{ij})_{4\times4}\quad \mbox{with } G_{ij}=M^{*}_{i}GM_{j}, i,j=1,2,3,4. $$
(3.13)

Then substitute (3.12) into (3.11). By [13, Theorem 3.1] we obtain that equation (3.11) is consistent if and only if

$$ \begin{gathered} Y^{*}_{1}Y_{1}\Lambda+\Lambda^{*}Y^{*}_{2}Y_{2}= \Lambda^{*}Y^{*}_{1}Y_{1}+Y^{*}_{2}Y_{2} \Lambda,\\ G_{13}=0,\qquad G_{14}=0,\qquad G_{24}=0,\qquad G_{34}=0,\qquad G_{44}=0.\end{gathered} $$
(3.14)

Moreover, its general solution can be expressed as

$$\begin{aligned}& \widehat{S}_{1}=U_{1}\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} G_{11} & G_{12}\Lambda_{1}^{-1} & X_{13}\\ \Lambda_{1}^{-1}G^{*}_{12} & \Lambda_{1}^{-1}(G_{22}-\Lambda_{2}Y_{22}\Lambda _{2})\Lambda_{1}^{-1} & X_{23}\\ X^{*}_{13} & X^{*}_{23} & X_{33} \end{array}\displaystyle \right ]U^{*}_{1}, \end{aligned}$$
(3.15)
$$\begin{aligned}& \widehat{S}_{2}=U_{2}\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} Y_{11} & Y_{12} & Y_{13}\\ Y^{*}_{12} & Y_{22} & \Lambda_{2}^{-1}G_{23}\\ Y^{*}_{13} & G^{*}_{23}\Lambda_{2}^{-1} & G_{33} \end{array}\displaystyle \right ]U^{*}_{2}, \end{aligned}$$
(3.16)

where \(X_{33}\), \(Y_{11}\), and \(Y_{22}\) are arbitrary Hermitian matrices, \(X_{13}\), \(X_{23}\), \(Y_{12}\), and \(Y_{13}\) are arbitrary matrices.

Then substituting (3.15) and (3.16) into (3.7) and (3.9) yields

$$ \left\{ \textstyle\begin{array}{l} A_{11}=Y_{1}\Lambda{\mathscr{L}}_{Y_{2}}{(Y_{1}{\mathscr{L}}_{Y_{2}})}^{\dagger}+{({\mathscr{L}}_{Y_{2}}Y^{*}_{1})}^{\dagger}{\mathscr{L}}_{Y_{2}} \Lambda^{*}Y^{*}_{1}{\mathscr{R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}} +{\mathscr{R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}\widehat{S}_{1}{\mathscr {R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}},\\ A_{22}=Y_{2}\Lambda{\mathscr{L}}_{Y_{1}}{(Y_{2}{\mathscr{L}}_{Y_{1}})}^{\dagger}+{({\mathscr{L}}_{Y_{1}}Y^{*}_{2})}^{\dagger}{\mathscr{L}}_{Y_{1}} \Lambda^{*}Y^{*}_{2}{\mathscr{R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}} +{\mathscr{R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}\widehat{S}_{2}{\mathscr {R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}. \end{array}\displaystyle \right. $$
(3.17)

From (3.4), (3.17), and Lemma 3, we get

$$ \begin{aligned}[b] A_{12} ={}& {\bigl(Y^{*}_{1} \bigr)}^{\dagger}\bigl(Y^{*}_{2}A_{22}- \Lambda^{*}Y^{*}_{2}\bigr)+{\mathscr {R}}_{Y_{1}}(Y_{1} \Lambda-A_{11}Y_{1}){\bigl(Y^{*}_{2} \bigr)}^{\dagger}+ {\mathscr {R}}_{Y_{1}}R{\mathscr{R}}_{Y_{2}} \\ ={}& {\bigl(Y^{\dagger}_{1}\bigr)}^{*}Y^{*}_{2}Y_{2} \Lambda{\mathscr{L}}_{Y_{1}}{(Y_{2}{\mathscr {L}}_{Y_{1}})}^{\dagger}+ {\bigl(Y^{\dagger}_{1}\bigr)}^{*}Y^{*}_{2} {\bigl({\mathscr {L}}_{Y_{1}}Y_{2}^{*}\bigr)}^{\dagger}{\mathscr{L}}_{Y_{1}} \Lambda^{*}Y^{*}_{2}{\mathscr {R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}} \\ &-{\mathscr{R}}_{Y_{1}}{\bigl({\mathscr{L}}_{Y_{2}}Y^{*}_{1} \bigr)}^{\dagger}{\mathscr {L}}_{Y_{2}}\Lambda^{*}Y^{*}_{1}{ \mathscr{R}}_{Y_{1}{\mathscr {L}}_{Y_{2}}}Y_{1}{Y_{2}}^{\dagger}-{ \bigl(Y^{\dagger}_{1}\bigr)}^{*}\Lambda^{*}Y^{*}_{2} \\ &+{\bigl(Y^{\dagger}_{1}\bigr)}^{*}Y^{*}_{2}{ \mathscr{R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}\widehat {S}_{2}{\mathscr{R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}- {\mathscr{R}}_{Y_{1}}{\mathscr{R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}\widehat {S}_{1}{\mathscr{R}}_{Y_{1}{\mathscr{L}}_{Y_{2}}}Y_{1}{Y_{2}}^{\dagger}\\ &+{\mathscr{R}}_{Y_{1}}R{\mathscr{R}}_{Y_{2}} ,\end{aligned} $$
(3.18)

where \(R \in{\mathbb{C}}^{k\times k}\) is an arbitrary matrix.

Based on the above discussion, we can conclude the following result to solve Problem 1.

Theorem 1

Given \(Y \in{\mathbb{C}}^{n\times m}\) and \(\Lambda\in {\mathbb{C}}^{m\times m}\) as described in Problem 1. Let \(U^{*}Y\), G, the GSVD of the matrix pair \(({\mathscr{R}}_{Y_{1}{\mathscr {L}}_{Y_{2}}}Y_{1}, {\mathscr{R}}_{Y_{2}{\mathscr{L}}_{Y_{1}}}Y_{2})\) and \({(M^{*})}^{-1}G{M}^{-1}\) be given by (3.3), (3.10), (3.12), and (3.13), respectively. Then Problem 1 is solvable(i.e., \(\mathcal{M}(Y,\Lambda)\neq\emptyset\)) in the set \({\mathcal{NS}}^{n\times n}(J)\) if and only if conditions (3.6), (3.8), (3.14), and \(A_{11}A_{12}=A_{12}A_{22}\) hold. Moreover, in this case, the general solution can be expressed as

$$A=U\left [ \textstyle\begin{array}{c@{\quad}c} A_{11} & A_{12}\\ -A^{*}_{12} & A_{22} \end{array}\displaystyle \right ]U^{*}, $$

where \(A_{11}\), \(A_{22}\), and \(A_{12}\) are given by (3.17) and (3.18), respectively. In the matrices \(A_{11}\), \(A_{22}\), and \(A_{12}\), \(\widehat{S}_{1}\) and \(\widehat{S}_{2}\) are described in (3.15) and (3.16), respectively, where \(X_{33}\), \(Y_{11}\), and \(Y_{22}\) are arbitrary Hermitian matrix blocks, \(X_{13}\), \(X_{23}\), \(Y_{12}\), \(Y_{13}\), and R are arbitrary matrix blocks.

4 Conclusions

In this paper, we have obtained the necessary and sufficient conditions of the inverse eigenvalue problem for normal skew J-Hamiltonian matrices. Furthermore, a solvable general representation is presented. We can also use the same method to solve the inverse eigenvalue problem for a normal J-Hamiltonian matrix.