Hadamard variation formula

Short description: Formula in matrix theory

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 $n\times n$ Hermitian matrices with all eigenvalues distinct.

Let $A=A(t)$ be a path in the space. Let $u_i,{\lambda }_i$ be its eigenpairs.

Hadamard variation formula (Tao 2012, pp. 48–49) — If $A(t)$ is first-differentiable, then ${\dot{\lambda }}_i=u_i^{*}\dot{A}{u}_i$

If $A(t)$ is second-differentiable, then ${\ddot{\lambda }}_i=u_i^{*}\ddot{A}{u}_i+2\sum _{j\ne i}\frac{{\left|u_i^{*}\dot{A}{u}_j\right|}^2}{{\lambda }_i-{\lambda }_j}$

Proof

Since $u_i^{*}{u}_i=1$ does not change with time, taking the derivative, we find that $⟨{\dot{u}}_i,u_i⟩$ is purely imaginary. Now, this is due to a unitary ambiguity in the choice of $u_i(t)$. Namely, for any first-differentiable $\theta (t)$, we can pick $v_i(t):=e^{i\theta (t)}{u}_i(t)$ instead. In that case, we have $$⟨{\dot{v}}_i,v_i⟩=⟨{\dot{u}}_i,u_i⟩-i\dot{\theta }$$ so picking $\theta$ such that $\dot{\theta }=-i⟨{\dot{u}}_i,u_i⟩$, we have $⟨{\dot{v}}_i,v_i⟩=0$. Thus, WLOG, we assume that $⟨{\dot{u}}_i,u_i⟩=0$.

Take derivative of $A{u}_i={\lambda }_i{u}_i$, $$\dot{A}{u}_i+A{\dot{u}}_i={\dot{\lambda }}_i{u}_i+{\lambda }_i{\dot{u}}_i$$ Now take inner product with $u_i$.

Taking derivative of $⟨{\dot{u}}_i,u_i⟩=0$, we get $$⟨{\ddot{u}}_i,u_i⟩=⟨u_i,{\ddot{u}}_i⟩=-⟨{\dot{u}}_i,{\dot{u}}_i⟩$$ and all terms are real.

Take derivative of $\dot{A}{u}_i+A{\dot{u}}_i={\dot{\lambda }}_i{u}_i+{\lambda }_i{\dot{u}}_i$, then multiply by $u_i^{*}$, and simplify by $u_i^{*}{\dot{u}}_i=0$, $u_i^{*}A={\lambda }_i{u}_i^{*}$, we get $$u_i^{*}\ddot{A}{u}_i+2u_i^{*}\dot{A}{\dot{u}}_i={\ddot{\lambda }}_i$$ - Expand ${\dot{u}}_i$ in the eigenbasis $\left\{u_j\right\}$ as ${\dot{u}}_i=\sum _{j\ne i}{c}_{ij}{u}_j$. Take derivative of $A{u}_i={\lambda }_i{u}_i$, and multiply by $u_j^{*}A$, we obtain $c_{ij}=-\frac{u_j^{*}\dot{A}{u}_i}{{\lambda }_i-{\lambda }_j}$.

Higher order generalizations appeared in (Tao Vu).

References

• Tao, Terence; Vu, Van (2011). "Random matrices: Universality of local eigenvalue statistics" (in en). Acta Mathematica 206 (1): 127–204. doi:10.1007/s11511-011-0061-3. ISSN 0001-5962. http://projecteuclid.org/euclid.acta/1485892530. 
• Tao, Terence (2012). Topics in random matrix theory. Graduate studies in mathematics. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-7430-1.