# Hadamard variation formula

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 n×n{\textstyle n\times n} Hermitian matrices with all eigenvalues distinct.
Let A=A(t){\textstyle A=A(t)} be a path in the space. Let u$_i$, λ$_i${\textstyle u_{i}, \lambda _{i}} be its eigenpairs.
Hadamard variation formula (Tao 2012, pp. 48–49) — If A(t){\textstyle A(t)} is first-differentiable, then λ$^˙$$_i$=u$_i$$^∗$A$^˙$u$_i${\textstyle {\dot {\lambda }}_{i}=u_{i}^{*}{\dot {A}}u_{i}}
If A(t){\textstyle A(t)} is second-differentiable, then λ$^¨$$_i$=u$_i$$^∗$A$^¨$u$_i$+2∑$_{j≠i}$ |u$_i$$^∗$A$^˙$u$_j$|$^2$/λ$_i$−λ$_j$ {\textstyle {\ddot {\lambda }}_{i}=u_{i}^{*}{\ddot {A}}u_{i}+2\sum _{j\neq i}{\frac {\left|u_{i}^{*}{\dot {A}}u_{j}\right|^{2}}{\lambda _{i}-\lambda _{j}}}}
Since u$_i$$^∗$u$_i$=1{\textstyle u_{i}^{*}u_{i}=1} does not change with time, taking the derivative, we find that ⟨u$^˙$$_i$, u$_i$⟩{\textstyle \langle {\dot {u}}_{i}, u_{i}\rangle } is purely imaginary. Now, this is due to a unitary ambiguity in the choice of u$_i$(t){\textstyle u_{i}(t)}. Namely, for any first-differentiable θ(t){\textstyle \theta (t)}, we can pick v$_i$(t):=e$^{iθ(t)}$u$_i$(t){\textstyle v_{i}(t):=e^{i\theta (t)}u_{i}(t)} instead. In that case, we have $\langle {\dot {v}}_{i},v_{i}\rangle =\langle {\dot {u}}_{i},u_{i}\rangle -i{\dot {\theta }}$ so picking θ{\textstyle \theta } such that θ$^˙$=−i⟨u$^˙$$_i$, u$_i$⟩{\textstyle {\dot {\theta }}=-i\langle {\dot {u}}_{i}, u_{i}\rangle }, we have ⟨v$^˙$$_i$, v$_i$⟩=0{\textstyle \langle {\dot {v}}_{i}, v_{i}\rangle =0}. Thus, WLOG, we assume that ⟨u$^˙$$_i$, u$_i$⟩=0{\textstyle \langle {\dot {u}}_{i}, u_{i}\rangle =0}.
Take derivative of Au$_i$=λ$_i$u$_i${\textstyle Au_{i}=\lambda _{i}u_{i}}, ${\dot {A}}u_{i}+A{\dot {u}}_{i}={\dot {\lambda }}_{i}u_{i}+\lambda _{i}{\dot {u}}_{i}$ Now take inner product with u$_i${\textstyle u_{i}}.
Taking derivative of ⟨u$^˙$$_i$, u$_i$⟩=0{\textstyle \langle {\dot {u}}_{i}, u_{i}\rangle =0}, we get $\langle {\ddot {u}}_{i},u_{i}\rangle =\langle u_{i},{\ddot {u}}_{i}\rangle =-\langle {\dot {u}}_{i},{\dot {u}}_{i}\rangle$ and all terms are real.
Take derivative of A$^˙$u$_i$+Au$^˙$$_i$=λ$^˙$$_i$u$_i$+λ$_i$u$^˙$$_i${\textstyle {\dot {A}}u_{i}+A{\dot {u}}_{i}={\dot {\lambda }}_{i}u_{i}+\lambda _{i}{\dot {u}}_{i}}, then multiply by u$_i$$^∗${\textstyle u_{i}^{*}}, and simplify by u$_i$$^∗$u$^˙$$_i$=0{\textstyle u_{i}^{*}{\dot {u}}_{i}=0}, u$_i$$^∗$A=λ$_i$u$_i$$^∗${\textstyle u_{i}^{*}A=\lambda _{i}u_{i}^{*}}, we get $u_{i}^{*}{\ddot {A}}u_{i}+2u_{i}^{*}{\dot {A}}{\dot {u}}_{i}={\ddot {\lambda }}_{i}$ - Expand u$^˙$$_i${\textstyle {\dot {u}}_{i}} in the eigenbasis {u$_j$}{\textstyle \left\{u_{j}\right\}} as u$^˙$$_i$=∑$_{j≠i}$c$_{ij}$u$_j${\textstyle {\dot {u}}_{i}=\sum _{j\neq i}c_{ij}u_{j}}. Take derivative of Au$_i$=λ$_i$u$_i${\textstyle Au_{i}=\lambda _{i}u_{i}}, and multiply by u$_j$$^∗$A{\textstyle u_{j}^{*}A}, we obtain c$_{ij}$=−u$_j$$^∗$A$^˙$u$_i$/λ$_i$−λ$_j$ {\textstyle c_{ij}=-{\frac {u_{j}^{*}{\dot {A}}u_{i}}{\lambda _{i}-\lambda _{j}}}}.
Higher order generalizations appeared in (Tao & Vu 2011).