Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix? [closed]

Question

I have a real symmetric $3\times3$ matrix $\mathbf{M}(\mathbf{r}$) which depends on $\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field $e_i(\mathbf{r})$ over $\mathbb{R}^3$. Assume these functions are differential.

A general derivative $\partial e_i(\mathbf{r}) = \mathbf{v}_i^T.\partial\mathbf{M}(\mathbf{r}).\mathbf{v_i}$ where $\mathbf{v}_i$ is the eigenvector corresponding to eigenvalue $e_i$ and $\partial\mathbf{M}(\mathbf{r})$ is the matrix of derivatives of the elements.

Using the above, $\vec{\nabla}e_i(\mathbf{r})$ is straightforward to determine, but $\Delta e_i(\mathbf{r}) = \vec{\nabla}.\vec{\nabla}e_i(\mathbf{r})$ is proving to be a bit more challenging.

I'd be happy with any tips or references.

Answer 1

The standard reference for this is Kato's Perturbation Theory for Linear Operators (the finite-dimensional case is analyzed in detail in Chapter 1; the self-adjoint [symmetric] case is easier, of course. I think it's on page 72, but around there, anyhow.