Noncommutative Poisson structure and invariants of matrices

We introduce a novel approach that employs techniques from noncommutative Poisson geometry to comprehend the algebra of invariants of two \(n\times n\) matrices. We entirely solve the open problem of computing the algebra of invariants of two \(4 \times 4\) matrices. As an application, we derive the complete description of the invariant commuting variety of \(4 \times 4\) matrices and the fourth Calogero-Moser space.