Consider a discrete-time LTI system described by the state evolution relation \begin{equation} \label{eq:basic-evolution} x_{t+1} = Ax_t + Bu_t, \end{equation} where \(x_t\), \(x_{t+1}\in \mathbb{C}^{n}\), \(u_t\in \mathbb{C}^m\), and \(A\) and \(B\) are complex matrices of appropriate size. Suppose we wish to invoke a change of state variable by the linear transformation \(z = Tx\) and eliminate \(x\); in the case where \(T\) is invertible, a simple manipulation yields the transformed state evolution relation \begin{equation} \label{eq:transformed-evolution} z_{t+1} = T^{-1}ATz_t + T^{-1}Bu_t. \end{equation} Despite having different state variables and evolving according to a different linear transformation, (??) and (??) are effectively describing the “same” dynamical system, since one can compute \(z_t\) either by evolving \(z_0\) for \(t\) time steps using (??), or by evolving \(x_0\) for \(t\) time steps using (??) and applying the transformation \(z_t = Tx_t\) only at the final step. In other words, applying the change of variable \(z=Tx\) preserves the essential dynamical structure of the system.

This type of structure-preserving transformation is a crucial analytical technique in linear systems theory. For example, it plays a key role in showing that the stability of the system is determined by the eigenvalues of \(A\). In this case, we choose a \(T\) that diagonalizes¹ \(A\), so that the dynamics of each component of \(z\) is decoupled from the rest. The transformed system is trivial to analyze; and since it is “the same system” despite its simplicity, our analysis of the diagonalized system applies to the original system as well.

The transformation \(z=Tx\) is a special example of a general class of structure-preserving transformations between dynamical systems called dynamorphisms. In addition to their importance in analysis, dynamorphisms are important because they serve as morphisms in the category of dynamical systems. Here is how dynamorphisms arise in the categorical setting. Wackywoo!