Question

Show that the systems \(\dot{z}_{1}=x_{1}, \dot{x}_{2}=x_{2}\), and \(\dot{x}_{1}=x_{1}+x_{2}, \dot{x}_{2}=x_{2}\) are linearly inequivalent, even though they have the same eigenvalues.

Short answer

Answer: The given systems are linearly inequivalent because they have different eigenvalues.

Step-by-step solution

Write the systems in matrix form

First, we write the given systems in the matrix form \(\dot{X}=AX\), where \(X\) and \(A\) represent the column vector of variables and the system matrix, respectively: For the first system, \(\dot{z}_{1}=x_{1}\) and \(\dot{x}_{2}=x_{2}\): \(X=\begin{bmatrix} z_{1}\\ x_{2} \end{bmatrix}\) and \(A=\begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix}\) For the second system, \(\dot{x}_{1}=x_{1}+x_{2}\) and \(\dot{x}_{2}=x_{2}\): \(X=\begin{bmatrix} x_{1}\\ x_{2} \end{bmatrix}\) and \(A=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)

Find the eigenvalues of the matrices

We find the eigenvalues of both matrices A by solving the characteristic equation \(|A-\lambda I|=0\): For the first matrix: \(|\begin{bmatrix} -\lambda & 1 \\ 0 & 1-\lambda \end{bmatrix}|=(-\lambda)(1-\lambda)-0*1=\lambda^{2}-\lambda=0\) The eigenvalues of the first matrix are \(\lambda_1=1\) and \(\lambda_2=0\). For the second matrix: \(|\begin{bmatrix} 1-\lambda & 1 \\ 0 & 1-\lambda \end{bmatrix}|=(1-\lambda)^{2}-0*1=(1-\lambda)^{2}=0\) The eigenvalues of the second matrix are \(\lambda_1=1\) and \(\lambda_2=1\). The two sets of eigenvalues are not the same, so we can safely say the two systems are linearly inequivalent. There is no need to check for similarity between the matrices.

Key concepts

Eigenvalues

Eigenvalues are crucial in understanding the behavior of differential equations, especially when examining linear systems. In simple terms, an eigenvalue describes the factors by which a transformation stretches or shrinks space in its direction. When you have a square matrix, eigenvalues can be determined by solving the characteristic equation. These values help in analyzing the stability and behavior of solutions to a system of differential equations.

For each system of equations, eigenvalues indicate how solutions evolve over time. Systems with different eigenvalues often have different dynamic behaviors. This means that even systems with identical eigenvalues can behave similarly or differently, depending on other properties such as their eigenvectors. In this problem, determining the eigenvalues was the first step to understanding how each system behaves.

Matrix Form

Matrix form is a way to express a set of linear differential equations as a single matrix equation. This format simplifies the analysis and allows mathematicians to use tools from linear algebra to solve or understand the system’s behavior.

In the problem, transforming systems into matrix form involved creating matrices that represent coefficients of the system and vectors for its variables. Each linear system can be denoted as \( \dot{X} = AX \), where \( X \) is the state vector, and \( A \) is the matrix containing coefficients. By arranging the systems like this, properties such as eigenvalues and eigenvectors become more accessible, providing insights into the systems' dynamics.

This approach particularly benefits from clarity and conciseness, allowing quick visual assessment of how systems differ or resemble each other.

Characteristic Equation

The characteristic equation is an essential tool for finding eigenvalues of a matrix. It establishes a polynomial whose roots yield the eigenvalues of the system’s matrix. The general form is derived from the matrix equation \( |A - \lambda I| = 0 \), where \( A \) is the system matrix and \( \lambda \) represents the eigenvalues.

Solving the characteristic equation involves calculating the determinant of \( A - \lambda I \), where \( I \) is the identity matrix. This determinant gives us the characteristic polynomial, and solving for \( \lambda \) results in eigenvalues.

In the problem, computing the characteristic equation for each system helped us find their eigenvalues, crucial to comparing the systems. Despite the initial problem suggesting similar eigenvalues for the systems, direct computation clarified their differences.

Linear Equivalence

Linear equivalence between two systems of differential equations implies that they can be transformed into one another using a linear transformation. This reflects shared dynamic characteristics like solution behavior over time. Two systems share linear equivalence if they have both identical eigenvalues and eigenvectors, indicating similar transformations in space.

In this exercise, determining linear equivalence involved comparing eigenvalues of the systems’ matrices. Since the discovered eigenvalues differed for the tested systems, the systems are linearly inequivalent. This means they exhibit fundamentally different dynamics in response to similar conditions.

Recognizing linear inequivalence is important in categorizing system behavior and ensuring that any predictions or analyses consider these fundamental distinctions in behavior.