Question

Suppose the real parts of all the eigenvalues of \(A\) are negative. Prove that a phase flow \(g^{t}\) of Eq. (1) decreases volumes \((t>0)\).

Short answer

Question: Prove that if all the eigenvalues of a matrix A have negative real parts, the phase flow of a differential equation decreases volumes (t > 0). Answer: This statement is true because the determinant of the matrix exponential (which represents the phase flow) is equal to the exponential of the trace of the matrix. If all eigenvalues have negative real parts, the trace will be negative, making the determinant less than 1 and leading to a decrease in volume for t > 0.

Step-by-step solution

Recall the definition of phase flow

For a given differential equation \(\dot{x} = Ax\), the phase flow \(g^t\) maps the initial state \(x(0)\) to the state \(x(t)\) at time \(t\). Mathematically, we can represent this as \(x(t) = g^t(x(0))\).

Understand the relation between eigenvalues and phase flow

Eigenvalues of the matrix A determine the behavior of the phase flow. If all eigenvalues have negative real parts, then the phase flow will contract along each eigenvector.

Relate phase flow to the determinant of the matrix exponential

The phase flow \(g^t\) can be expressed in terms of the matrix exponential: \(x(t) = e^{At}x(0)\). Now, to study how volumes change under the phase flow, we look at the determinant of the matrix exponential, that is, \(\det(e^{At})\).

Use a property of determinant and eigenvalues to find \(\det(e^{At})\)

We know that the determinant of the matrix exponential is equal to the exponential of the trace of the matrix: \(\det(e^{At}) = e^{\text{tr}(At)}\). The trace of a matrix is equal to the sum of its eigenvalues. So, \(\text{tr}(At) = \sum_i (\lambda_i t)\), where \(\lambda_i\) are the eigenvalues of matrix A. Since all the eigenvalues have negative real parts, then \(\text{tr}(At) < 0\). Therefore, \(\det(e^{At}) < 1\).

Prove that phase flow decreases volumes

Given that \(\det(e^{At}) < 1\), the volume contraction can be seen by considering the volume element represented by the matrix exponential \(e^{At}\). The volume change is given by the determinant of this matrix, and since its determinant is less than 1, the volume will decrease for \(t > 0\). Thus, we have proven that if all the eigenvalues of A have negative real parts, the phase flow decreases volumes \((t > 0)\).

Key concepts

Eigenvalues and Phase Flow

Understanding the connection between eigenvalues and phase flow is crucial when studying differential equations. In the context of a linear system defined by \( \dot{x} = Ax \), the eigenvalues of the coefficient matrix \( A \) dictate the response of the system over time. When eigenvalues have negative real parts, they indicate that the system has a tendency to converge to a stable state as time progresses. Imagine a phase portrait, a graphical representation of trajectories in a dynamic system. Each eigenvalue correlates to a direction or dimension in this portrait.

With negative eigenvalues, the trajectories of the system will spiral inward, converging towards a point, known as an attractor. This behavior results in the phase flow, the evolution of the system's state over time, gradually shrinking in volume. This contraction can be visualized as if the system's state space is made of elastic material that's being pulled tighter and tighter, leading to a squeeze in volumes as time passes. This reflects a fundamental property in dynamical systems where the phase space's geometry is transformed by the system's evolution.

Matrix Exponential and Volume Change

The matrix exponential plays a key role when analyzing how the state of a dynamical system evolves over time. For a linear time-invariant system, the matrix exponential \(e^{At}\) acts as an operator that maps the initial conditions to their state after time \(t\). Intuitively, you can think of the matrix exponential as describing all possible states the system can transition into, continuously compounded over time.

To appreciate the impact of the matrix exponential on the volume of phase space, imagine a multi-dimensional shape representing the initial conditions. Over time, each point on this shape is moved and possibly distorted by the action of \(e^{At}\). The determinant of this matrix can then be interpreted as a scaling factor: how much the volume of this shape changes after time \(t\). A determinant greater than one signifies expansion, while a determinant less than one indicates contraction.

When eigenvalues of \(A\) have negative real parts, as established, the trace of the matrix (the sum of eigenvalues) will be negative. Consequently, the determinant of \(e^{At}\) will be less than one, signifying a decrease in volume as the system evolves—validating the concept through a mathematical lens.

Determinant and Volume Contraction in Dynamical Systems

Delving deeper into how the determinant influences dynamical systems: the determinant of a matrix can be seen as a measure of how much an operation represented by the matrix scales the volume in the space it operates in. If we have a determinant of one, the operation preserves volume; if it's less than one, it contracts volume; and if it's greater than one, it expands volume.

For dynamical systems governed by \( \dot{x} = Ax \), the evolution of volumes in the phase space can be assessed by looking at the determinant of \(e^{At}\). The rule here is straightforward - if the determinant is less than one, the corresponding transformation compresses the volume. This principle shows how the inherent properties of a linear system—captured by the matrix \(A\)—manifest in geometric terms. When all eigenvalues of \(A\) hold negative real parts, \(\text{tr}(At) < 0\) leading to a determinant of \(e^{At}\) that is less than one, enforcing a systematic contraction of volumes for all \( t > 0 \). This contraction is not just an abstract concept but has real implications for predicting the behavior of physical and theoretical systems described by such differential equations.