Unravelling the Mathematical Marvels: Exploring the Inertia of Pattern-Based Matrices and Their Behaviour under Operator Concave and Convex Function

This article explores the inertia properties of pattern-based matrices and their responses to operator concave ( ) and convex (  functions. Pattern-based matrices, defined by structured element arrangements, offer rich mathematical terrain. We investigate the interplay between matrix patterns and inertia, crucial in optimization and eigenvalue theory. Considering an  matrix  with elements , we analyze eigenvalues  ) and singular values  ) distributions across various patterns. In addition, we analyse the role of operator function in matrix trace modification, which affect spectral characteristics and geometric properties. By means of convex analysis and numerical experiments we unveil the inertia dynamics under operator functions to provide knowledge relevant in data science, signal processing and optimization.   Keywords:  Pattern-based matrices; Inertia dynamics; Operator functions; Eigenvalues; Singular values. This article explores the inertia properties of pattern-based matrices and their responses to operator concave ( ) and convex (  functions. Pattern-based matrices, defined by structured element arrangements, offer rich mathematical terrain. We investigate the interplay between matrix patterns and inertia, crucial in optimization and eigenvalue theory. Considering an  matrix  with elements , we analyze eigenvalues  ) and singular values  ) distributions across various patterns. In addition, we analyse the role of operator function in matrix trace modification, which affect spectral characteristics and geometric properties. By means of convex analysis and numerical experiments we unveil the inertia dynamics under operator functions to provide knowledge relevant in data science, signal processing and optimization.