# The Nearest Hermitian Inverse Eigenvalue Problem Solution with Respect to the 2-Norm

## TL;DR

This paper presents a method to find a Hermitian matrix with specified eigenvalues that is closest to a given Hermitian matrix in the 2-norm, with applications demonstrated on random matrices and images.

## Contribution

It introduces a solution to the inverse eigenvalue problem for Hermitian matrices minimizing the 2-norm distance to a given matrix, including proofs and practical tests.

## Key findings

- The method effectively finds Hermitian matrices with desired eigenvalues.
- Eigenvalue corrections exhibit a smoothing effect on images.
- The approach is validated through experiments with random matrices and images.

## Abstract

Assume that the eigenvalues of a finite hermitian linear operator have been deduced accurately but the linear operator itself could not be determined with precision. Given a set of eigenvalues $\lambda$ and a hermitian matrix $M$, this paper will explain, with proofs, how to find a hermitian matrix $A$ with the desired eigenvalues $\lambda$ that is as close as possible to the given operator $M$ according to the operator 2-norm metric. Furthermore the effects of this solution are put to a test using random matrices and grayscale images which evidently show the smoothing property of eigenvalue corrections.