Question

In Exercises \(29-32\) find vectors \(\mathbf{U}\) and \(\mathbf{V}\) parallel to the axes of symmetry of the trajectories, and plot some typical trajectories. $$ \text { C/G } \mathbf{y}^{\prime}=\left[\begin{array}{rr} -3 & -15 \\ 3 & 3 \end{array}\right] \mathbf{y} $$

Short answer

Answer: The axes of symmetry of the trajectories for the given matrix A can be determined by the eigenvectors U and V: $$ \mathbf{U} = \begin{bmatrix} 5 \\ 1 \end{bmatrix} \quad \text{ and } \quad \mathbf{V} = \begin{bmatrix} 4 \\ -1 \end{bmatrix} $$

Step-by-step solution

Find the eigenvalues

First we need to find the eigenvalues of the given matrix A: $$ A = \begin{bmatrix} -3 & -15 \\ 3 & 3 \end{bmatrix} $$ To find the eigenvalues, we need to solve the following matrix equation: $$ \det(A - \lambda I) = 0 $$ $$ \det\left(\begin{bmatrix} -3-\lambda & -15 \\ 3 & 3-\lambda \end{bmatrix}\right)=0 $$ Calculate the determinant and find the roots of the polynomial equation: $$ (-3-\lambda)(3-\lambda) - (-15)(3) = 0 \\ \Rightarrow \lambda^2 - 9 = 0 $$ From here, we can find the eigenvalues as λ₁ = 3 and λ₂ = -3.

Find the eigenvectors

Now that we have the eigenvalues, we can find the corresponding eigenvectors. To find the eigenvectors, we need to solve the following equations for each eigenvalue: For λ₁ = 3, $$ (A - 3I)\mathbf{v} = 0 \\ \Rightarrow \begin{bmatrix} -6 & -15 \\ 3 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $$ Row reduce the augmented matrix: $$ \begin{bmatrix} -6 & -15 & 0 \\ 3 & 0 & 0 \end{bmatrix} $$ to obtain: $$ \begin{bmatrix} 1 & -5 & 0 \\ 0 & -15 & 0 \end{bmatrix} $$ The eigenvector corresponding to λ₁ = 3 is: $$ \mathbf{v_1} = \begin{bmatrix} 5 \\ 1 \end{bmatrix} $$ For λ₂ = -3, $$ (A + 3I)\mathbf{v} = 0 \\ \Rightarrow \begin{bmatrix} 0 & -15 \\ 3 & 6 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $$ Row reduce the augmented matrix: $$ \begin{bmatrix} 0 & -15 & 0 \\ 3 & 6 & 0 \end{bmatrix} $$ to obtain: $$ \begin{bmatrix} 3 & 6 & 0 \\ 0 & -15 & 0 \end{bmatrix} $$ The eigenvector corresponding to λ₂ = -3 is: $$ \mathbf{v_2} = \begin{bmatrix} 4 \\ -1 \end{bmatrix} $$

Confirm that U and V are parallel to the axes of symmetry

The eigenvectors found in step 2 are the vectors U and V for this problem. These eigenvectors determine the directions parallel to the axes of symmetry of the trajectories. $$ \mathbf{U} = \begin{bmatrix} 5 \\ 1 \end{bmatrix} \quad \text{ and } \quad \mathbf{V} = \begin{bmatrix} 4 \\ -1 \end{bmatrix} $$

Plot the trajectories

We can now plot some typical trajectories by following the directions of the vectors U and V. We will start by plotting U and V from the origin, to get a sense of the axes of symmetry. Typical trajectories can be plotted by taking the linear combinations of the eigenvectors. For example, if we have the basis of eigenvectors \(U\) and \(V\), then for some constants \(c_1\) and \(c_2\): $$ \mathbf{y}(t) = c_1\mathbf{U} + c_2\mathbf{V} $$ Using this representation and choosing different values for \(c_1\) and \(c_2\), we can generate different trajectories in the phase plane. Plotting these trajectories will show the typical behavior of the system and demonstrate the axes of symmetry.

Key concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts when dealing with systems of differential equations, particularly when the systems can be represented in matrix form. In the given exercise, the eigenvalues of a matrix A are calculated by determining when the determinant of the matrix minus \(\lambda I\) is equal to zero. The characteristic polynomial that results from this calculation gives us the eigenvalues. Once the eigenvalues \(\lambda_1\) and \(\lambda_2\) are found to be 3 and -3 respectively, eigenvectors can be determined. These eigenvectors are non-zero vectors that, when multiplied by matrix A, simply scale by the corresponding eigenvalue. To find the eigenvectors for each eigenvalue, one must solve \( (A - \lambda I)\mathbf{v} = 0 \) where A is our matrix and \(\lambda\) is the eigenvalue. In this problem, row reduction helps in simplifying the augmented matrix to find the eigenvectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\). Understanding these concepts is crucial because they represent the directions along which the behavior of the system is invariant, leading us to the next concept of phase plane analysis.

Matrix Algebra

Matrix algebra plays a role in the understanding of systems of differential equations as it provides a way to compactly represent and manipulate sets of equations. The system in this exercise is represented by a 2x2 matrix A, which helps in determining the system's behavior through its eigenvalues and eigenvectors. In matrix algebra, operations such as finding the determinant, performing row reductions, and multiplying vectors by matrices are essential. These operations allow us to transform the system into forms that are easier to analyze and understand. For example, row reduction in the exercise makes it possible to isolate the eigenvectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\). Additionally, matrix algebra is the bridge connecting linear equations to geometric interpretations, such as those depicted in phase plane analysis.

• Finding determinants to obtain characteristic polynomials
• Row reducing augmented matrices for eigenvectors
• Matrix-vector multiplication to confirm scaling properties

Phase Plane Analysis

Phase plane analysis is a graphical method to study systems of two first-order linear differential equations. It involves plotting the system's behavior over time, with each point in the phase plane representing the system's state at a given time. The trajectories in the phase plane indicate how the system evolves. In the exercise, eigenvectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\) provide the directions parallel to the axes of symmetry of these trajectories, which strongly influence the overall behavior of the system. Axes of symmetry identified by eigenvalues and eigenvectors divide the phase plane into regions with different qualitative behavior.

Trajectories and Symmetry

For instance, trajectories tend to move away from an unstable point and towards a stable one, with the eigenvectors pointing towards these directions. This method is a powerful tool for visualizing the stability and long-term behavior of a system.

System of Differential Equations

In the context of a system of differential equations, we look at equations that describe how quantities change over time relative to each other. The provided exercise involves a linear system of differential equations which can be depicted as \(\mathbf{y}^{\text{'}} = A\mathbf{y}\), where A is the system's coefficient matrix. Solving such systems often requires finding the eigenvalues and eigenvectors to understand how perturbations in different directions either grow or decay over time.

Understanding Patterns of Change

Knowing that the eigenvalues are 3 and -3, we can predict that trajectories will diverge along the eigenvector associated with the positive eigenvalue and converge along the eigenvector associated with the negative eigenvalue. The matrix algebra and phase plane analysis are tools that give us the ability to predict and explain these patterns of change, which are represented by trajectories on the phase plane.