The solvability conditions for the inverse problem of Re-nnd and Re-pd matrices on a linear manifold

Abstract

Let the linear manifold $\mathcal{S}$ be
$$
\mathcal{S}= \left\{A \in \mathbb{C}^{n\times n} \Bigg{|}
f(A)=\|AX_{1}-Z\|^{2}+\|X_{1}^{\ast}A-W^{\ast}\|^{2}=min,\forall X_{1}, Z,
W\in \mathbb{C}^{n\times k} \right\},
$$
where $\|.\|$ is Frobenius norm. We consider the following problem:\\
{\bf {Problem P.}} Given $Y_{2} \in \mathbb{C}^{n\times m},$ $X_{2} \in \mathbb{C}^{n\times p},$ $D \in \mathbb{C}^{m\times p},$
find $A \in \mathbb{C}_{\geq 0}^{n\times n} (\mathcal{S})$ $(\mathbb{C}_{> 0}^{n\times n} (\mathcal{S}))$ such that
$$
Y_{2}^{\ast}AX_{2}=D,
$$
where
$
{\mathbb{C}}_{\geq 0}^{n \times n}(\mathcal{S})= \left\{A \in \mathbb{C}^{n\times n} \Bigg{|}
\frac{1}{2}(A+A^\ast)\geq 0, \forall A \in \mathcal{S} \right\}
\left (
{\mathbb{C}}_{> 0}^{n \times n}(\mathcal{S})= \left\{A \in \mathbb{C}^{n\times n} \Bigg{|}
\frac{1}{2}(A+A^\ast)> 0, \forall A \in \mathcal{S} \right\}
\right ).$

\hspace{2pt}
In 2008, Dragana \cite{bi2} considered the Re-nnd solutions of $AXB=C$ with the constraints that $A, B$ are nonnegative definite. To reduce conservatism, Yuan and Zuo \cite{bi25} proposed the Re-nnd and Re-pd solutions to $AXB=D$ by using the Moore-Penrose generalized inverses and orthogonal projectors in 2015, which has no constraints on coefficient matrices.
We can see that there has been significant research on the Re-pd and Re-nnd solutions to $AXB=D$. However, investigations into its inverse problem are relatively scarce.
Particularly, when $Y_{2}^{\ast}=I, D=X_{2}\Lambda, \Lambda=\mbox{diag}(\lambda_{1}, \lambda_{2}, ...,\lambda_{p})$ or $X_{2}=I, D=\Omega Y_{2}^{\ast}, \Omega=\mbox{diag}(\omega_{1}, \omega_{2}, ...,\omega_{m})$, Problem P comes to an inverse eigenvalue problem of Re-nnd and Re-pd matrices on a linear manifold. Actually, the inverse eigenvalue problems, especially on linear manifolds, are widely applied in many applied sciences and engineering technologies, such as solid mechanics, structural vibration design, automatic control and system physical parameter identification.

\hspace{2pt}
In this paper, we consider a class of inverse problem for Re-nnd and Re-pd matrices on a linear manifold. Utilizing
variable substitution, the generalized inverses, and some matrix decompositions, the solvability conditions are obtained and the representations of the general solutions for Problem P are given.