Hurwitz Matrices

2026-01-14

Hurwitz Matrices are a type of square matrix which represent a linear dynamical system with stable dynamics.

Formulation

To formulate Hurwitz Matrices we first must understand what a linear dynamical system (LDS) is. Given a Matrix , a linear dynamical system involving real state variables can be described as follows:

Euler Integration

The following is the most naive way to simulate this

import numpy as np
def step_lds(x: np.array, A: np.array, dt: float):
	return x + (dt * A @ x)

This is a classic example of needing to be careful with implementing maths using Code, as even with a very small dt, the simulated behaviour of the system can differ wildly to the analytically correct behaviour

Behaviour

In order to understand the behaviour of an arbitrary LDS, we need to understand eigenvectors and eigenvalues.

Eigenvalues

Normally, when you apply an arbitrary matrix

to a vector

you get a whole new vector out

which bears no resemblance to the original input. Eigenvectors are the exception, wherein the action of the matrix on the vector is that of a scaling transformation:

This is very important to the behaviour of an LDS. If we suppose, that for some

, it's eigenvalue lies on the real line

and

, then

is going to be facing in exactly the same direction as

. Over time the vector will diverge. This is the opposite of what we call a stable LDS. Therefore, we require that all

eigenvalues have negative real part.