Hadamard variation formula

Consider the space of $n\backslash times\; n$ Hermitian matrices with all eigenvalues distinct.

Let $A=A(t)$ be a path in the space. Let $u\_i,\; \backslash lambda\_i$ be its eigenpairs.

{{Math theorem

| math_statement = If $A(t)$ is first-differentiable, then $\backslash dot\{\backslash lambda\}\_i=u\_i^*\; \backslash dot\{A\}\; u\_i$

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

| name = Hadamard variation formula

| note = {{harvnb|Tao|2012|pages=48–49}}

}}

{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=Proof}}

{{Math proof|title=Proof|proof=

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

Take derivative of $Au\_i\; =\; \backslash lambda\_i\; u\_i$, $$$$

\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 $\backslash langle\; \backslash dot\; u\_i,\; u\_i\backslash rangle\; =\; 0$, 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 $\backslash dot\{A\}\; u\_i+A\; \backslash dot\{u\}\_i=\backslash dot\{\backslash lambda\}\_i\; u\_i+\backslash lambda\_i\; \backslash dot\{u\}\_i$, then multiply by $u\_i^*$, and simplify by $u\_i^*\; \backslash dot\{u\}\_i=0$, $u\_i^*\; A=\backslash lambda\_i\; u\_i^*$, we get $$$$

u_i^* \ddot{A} u_i+2 u_i^* \dot{A} \dot{u}_i=\ddot{\lambda}_i

- Expand $\backslash dot\{u\}\_i$ in the eigenbasis $\backslash left\backslash \{u\_j\backslash right\backslash \}$ as $\backslash dot\{u\}\_i=\backslash sum\_\{j\; \backslash neq\; i\}\; c\_\{i\; j\}\; u\_j$. Take derivative of $Au\_i\; =\; \backslash lambda\_i\; u\_i$, and multiply by $u\_j^*A$, we obtain $c\_\{ij\}=-\backslash frac\{u\_j^*\; \backslash dot\{A\}\; u\_i\}\{\backslash lambda\_i-\backslash lambda\_j\}$.

}}{{hidden end}}Higher order generalizations appeared in {{Harvard citation|Tao|Vu|2011}}.

• {{Cite journal |last=Tao |first=Terence |last2=Vu |first2=Van |date=2011 |title=Random matrices: Universality of local eigenvalue statistics |url=http://projecteuclid.org/euclid.acta/1485892530 |journal=Acta Mathematica |language=en |volume=206 |issue=1 |pages=127–204 |doi=10.1007/s11511-011-0061-3 |issn=0001-5962|arxiv=0908.1982 }}
• {{Cite book |last=Tao |first=Terence |title=Topics in random matrix theory |date=2012 |publisher=American Mathematical Society |isbn=978-0-8218-7430-1 |series=Graduate studies in mathematics |location=Providence, R.I}}

Category:Matrix theory