In matrix theory, the Hadamard variation formula is a set of differential equations for how the eigenvalues of a time-varying Hermitian matrix with distinct eigenvalues change with time.
Statement
Consider the space of Hermitian matrices with all eigenvalues distinct.
Let be a path in the space. Let be its eigenpairs.
Since does not change with time, taking the derivative, we find that is purely imaginary. Now, this is due to a unitary ambiguity in the choice of . Namely, for any first-differentiable , we can pick instead. In that case, we have so picking such that , we have . Thus, WLOG, we assume that .
Take derivative of , Now take inner product with .
Taking derivative of , we get and all terms are real.
Take derivative of , then multiply by , and simplify by , , we get - Expand in the eigenbasis as . Take derivative of , and multiply by , we obtain . Higher order generalizations appeared in .
References