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 ${\textstyle n\times n}$ Hermitian matrices with all eigenvalues distinct.

Let ${\textstyle A=A(t)}$ be a path in the space. Let ${\textstyle u_{i},\lambda _{i}}$ be its eigenpairs.

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

If ${\textstyle A(t)}$ is second-differentiable, then ${\textstyle {\ddot {\lambda }}_{i}=u_{i}^{*}{\ddot {A}}u_{i}+2\sum _{j\neq i}{\frac {\left|u_{i}^{*}{\dot {A}}u_{j}\right|^{2}}{\lambda _{i}-\lambda _{j}}}}$

Proof

Proof

Since ${\textstyle u_{i}^{*}u_{i}=1}$ does not change with time, taking the derivative, we find that ${\textstyle \langle {\dot {u}}_{i},u_{i}\rangle }$ is purely imaginary. Now, this is due to a unitary ambiguity in the choice of ${\textstyle u_{i}(t)}$. Namely, for any first-differentiable ${\textstyle \theta (t)}$, we can pick ${\textstyle v_{i}(t):=e^{i\theta (t)}u_{i}(t)}$ 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 ${\textstyle \theta }$ such that ${\textstyle {\dot {\theta }}=-i\langle {\dot {u}}_{i},u_{i}\rangle }$, we have ${\textstyle \langle {\dot {v}}_{i},v_{i}\rangle =0}$. Thus, WLOG, we assume that ${\textstyle \langle {\dot {u}}_{i},u_{i}\rangle =0}$.

Take derivative of ${\textstyle Au_{i}=\lambda _{i}u_{i}}$, $${\dot {A}}u_{i}+A{\dot {u}}_{i}={\dot {\lambda }}_{i}u_{i}+\lambda _{i}{\dot {u}}_{i}$$ Now take inner product with ${\textstyle u_{i}}$.

Taking derivative of ${\textstyle \langle {\dot {u}}_{i},u_{i}\rangle =0}$, 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 ${\textstyle {\dot {A}}u_{i}+A{\dot {u}}_{i}={\dot {\lambda }}_{i}u_{i}+\lambda _{i}{\dot {u}}_{i}}$, then multiply by ${\textstyle u_{i}^{*}}$, and simplify by ${\textstyle u_{i}^{*}{\dot {u}}_{i}=0}$, ${\textstyle u_{i}^{*}A=\lambda _{i}u_{i}^{*}}$, we get $$u_{i}^{*}{\ddot {A}}u_{i}+2u_{i}^{*}{\dot {A}}{\dot {u}}_{i}={\ddot {\lambda }}_{i}$$ - Expand ${\textstyle {\dot {u}}_{i}}$ in the eigenbasis ${\textstyle \left\{u_{j}\right\}}$ as ${\textstyle {\dot {u}}_{i}=\sum _{j\neq i}c_{ij}u_{j}}$. Take derivative of ${\textstyle Au_{i}=\lambda _{i}u_{i}}$, and multiply by ${\textstyle u_{j}^{*}A}$, we obtain ${\textstyle c_{ij}=-{\frac {u_{j}^{*}{\dot {A}}u_{i}}{\lambda _{i}-\lambda _{j}}}}$.

Higher order generalizations appeared in (Tao & Vu 2011).

References

• Tao, Terence; Vu, Van (2011). "Random matrices: Universality of local eigenvalue statistics". Acta Mathematica. 206 (1): 127–204. arXiv:0908.1982. doi:10.1007/s11511-011-0061-3. ISSN 0001-5962.
• Tao, Terence (2012). Topics in random matrix theory. Graduate studies in mathematics. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-7430-1.