The spectral pinching of a matrix 
by a Hermitian matrix 
with spectral decomposition 
is
 |
where the sum is over the distinct eigenvalues of 
and the 
are its orthogonal
projections (Bhatia 1997). Thus pinching preserves each diagonal block associated
with an eigenspace of 
and removes the other blocks. Repeated eigenvalues retain entire blocks, rather than
only diagonal entries. The map 
is linear, preserves the matrix
trace, satisfies 
,
and produces a matrix that commutes with 
.
Dinh et al. (2026) represented the trace change caused by pinching using a positive measure on the plane. For Hermitian
matrices 
and 
,
there is a positive measure 
with compact support
such that
 |
for real 
and twice continuously differentiable real functions 
. The total mass is 
, where 
is the Frobenius norm
(Dinh et al. 2026). This representation yields the pinching refinement discussed
in the Bessis-Moussa-Villani conjecture.
Dinh et al. (2026) report using ChatGPT for mathematical calculations, exploration, and checking arguments, with the framework directed and the results verified by the authors.
