# Eigenvalue Dynamics

I was playing around with a model of ellipses evolving over time, and it occurred to me that one could interpret a positive vector as the eigenvalues of a positive definite matrix. We may then consider the dynamics of the eigenvalues.

Let $$x_t \in \mathcal{R}^n_{++}$$ represent the eigenvalues of a matrix $$C_t \in \mathcal{S}^n_{++}$$. Suppose $$x_t$$ evolves according to the dynamics

$x_{t+1} = c \circ (A \mathbin{\diamond} x_t) \circ (B \mathbin{\diamond} u_t),$

where $$\mathbin{\diamond}$$ denotes geometric matrix multiplication, $$c \in \mathcal{R}^n_{++}$$, $$A \in \mathcal{R}^{n \times n}$$, $$B \in \mathcal{R}^{n \times p}$$, and $$u_t \in \mathcal{R}^p_{++}$$ is a controller input.

We can then steer $$x_t$$ to maximize (or minimize) the volume of $$C_t$$, which is given by

$\mathrm{vol}(C_t) = \kappa_n \det\left(C_t^{-1}\right)^{1/2}$

or alternatively

$\mathrm{vol}(C_t) = \kappa_n \prod_{i=1}^n x_i^{-1/2}.$

Here $$\kappa_n > 0$$ is a dimension dependent constant.

I am very curious if there are any applications of this model!

Written on October 31, 2020