A Specific Case of a Family of Symmetric Matrices That Are Simultaneously Diagonalizable via Congruence

Abstract

This paper establishes a new condition for a specific family of symmetric matrices to be simultaneously diagonalizable via congruence. The paper considers a structured family of symmetric matrices satisfying a particular rank condition, which ensures the existence of a nonsingular congruence transformation that simultaneously diagonalizes all the initial matrices. Explicit examples are provided to illustrate the sharpness of our condition.