A Comprehensive Note on Submatrix Constraint for Inverse Eigenvalue Problem of Symmetric Matrix

A matrix $P\in {\mathbb{R}}^{n\times n}$ is said to be $(R,S)$ symmetric matrix if $RPS=P$, where $R,S$ be nontrivial involutions. The conditions for solving the inverse eigenvalue problem with leading principal submatrix constraints using a $(R,S)$ symmetric matrix are derived. Additionally, the existence, uniqueness, and expression of the $(R,S)$ symmetric matrix solution to the inverse eigenvalue problem's best approximation problem. The problem's best approximation solution is computed using an algorithm that is also provided. A numerical example is provided to demonstrate the algorithm's viability.