Derivative of eigenvectors of a symmetric matrix-valued function Given a real symmetric $3\times3$ matrix $\mathsf{A}_{ij}$ and its derivative (w.r.t. some parameter, let's call it time) $\dot{\mathsf{A}}_{ij}$, I want to measure/obtain the rotation (rate and direction) of the eigenvectors (the eigenvectors of a real symmetric matrix form an orthonormal matrix). How can this be done?
Edit Since the eigenvectors of a real symmetric matrix are mutually orthogonal, the change of the eigenvectors can only be an overall rotation. An infinitesimal rotation is uniquely determined by the rate $\boldsymbol{\omega}$ such that $\dot{\boldsymbol{x}}=\boldsymbol{\omega}\times\boldsymbol{x}$ for any vector $\boldsymbol{x}$. My question then becomes how to obtain $\boldsymbol{\omega}$.

Suppose that a given (differentiable) matrix-valued function $\mathrm A : \mathbb R \to \mbox{Sym}_n(\mathbb R)$, where $\mbox{Sym}_n(\mathbb R)$ denotes the set of $n \times n$ real symmetric matrices, does have a time-varying spectral decomposition
$$\mathrm A (t) = \mathrm V (t) \, \Lambda (t) \,\mathrm V^\top (t)$$
where the columns of orthogonal matrix $\mathrm V (t)$ and the diagonal entries of diagonal matrix $\Lambda (t)$ at a given $t$ are the (unit) eigenvectors and eigenvalues of $\mathrm A (t)$, respectively. 
Differentiating with respect to time, we obtain a nonlinear matrix differential equation in $\rm V$ and $\Lambda$
$$\dot{\mathrm A} (t) = \dot{\mathrm V} (t) \, \Lambda (t) \,\mathrm V^\top (t) + \mathrm V (t) \, \dot\Lambda (t) \,\mathrm V^\top (t) + \mathrm V (t) \, \Lambda (t) \,\dot{\mathrm V}^\top (t)$$
where $\dot{\mathrm A}$ serves as known input. From $\mathrm A (0)$, we obtain initial conditions $\mathrm V (0)$ and $\Lambda (0)$. We have:


*

*$\binom{n+1}{2}$ ordinary differential equations.

*$n^2$ (algebraic) quadratic equations (to ensure that $\mathrm V$ stays orthogonal).

*$n^2 + n = (n+1) n = 2\binom{n+1}{2}$ functions to determine.
Unfortunately, I do not know how to solve this matrix ODE. In fact, I am not even sure that a time-varying spectral decomposition of a symmetric matrix-valued function is actually legal.
