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