Question

In Exercises \(33-40\) find vectors \(\mathbf{U}\) and \(\mathbf{V}\) parallel to the axes of symmetry of the shadow trajectories, and plot a typical trajectory. $$ \text { C/G } \mathbf{y}^{\prime}=\left[\begin{array}{rr} -1 & -5 \\ 20 & -1 \end{array}\right] \mathbf{y} $$

Short answer

Question: Given a matrix C/G, find the vectors U and V parallel to the axes of symmetry of the shadow trajectories, and describe how to plot a typical trajectory. Answer: To find vectors U and V, follow these steps: 1. Compute the eigenvalues of the matrix C/G by solving the characteristic equation det(C/G - λI) = 0. 2. Solve the quadratic equation for eigenvalues using the quadratic formula. 3. Compute the eigenvectors for each eigenvalue by solving (C/G - λI)v = 0 for both eigenvalues. 4. Normalize the eigenvectors and denote them U and V. 5. Use a graphing tool to plot a typical shadow trajectory along with eigenvectors U and V to visualize the axes of symmetry.

Step-by-step solution

1. Compute the eigenvalues of the matrix C/G

First, we need to find the eigenvalues of the given matrix. The eigenvalues, \(\lambda\), are solutions to the characteristic equation, which is det(C/G - \(\lambda\)I) = 0, where I is the identity matrix. $$ \text{det}(C/G - \lambda I) = \begin{vmatrix} -1-\lambda & -5 \\ 20 & -1-\lambda \end{vmatrix} = (-1-\lambda)(-1-\lambda) - (-5)(20) = 0 $$ Now, we can solve for \(\lambda\): $$ \lambda^2 + 2\lambda - 1 - 100 = \lambda^2 + 2\lambda - 101 = 0 $$ To find the eigenvalues \(\lambda_1\) and \(\lambda_2\), we require to solve this quadratic equation.

2. Solve the quadratic equation for eigenvalues

The solutions to the quadratic equation can be found using the quadratic formula: $$ \lambda = \frac{-b \pm \sqrt {b^2 - 4ac}}{2a} $$ In our equation, \(a = 1\), \(b = 2\), and \(c = -101\). Plugging these values into the quadratic formula, we have: $$ \lambda = \frac{-2 \pm \sqrt{2^2 - 4(1)(-101)}}{2(1)} = \frac{-2 \pm \sqrt{408}}{2} $$ So the eigenvalues are: $$ \lambda_1 = \frac{-2 + \sqrt{408}}{2} \quad \text{and} \quad \lambda_2 = \frac{-2 - \sqrt{408}}{2} $$

3. Compute the eigenvectors for each eigenvalue

Now we need to find the corresponding eigenvectors, which are the axes of symmetry of the shadow trajectories. We will find the eigenvectors for each eigenvalue in the equation (C/G - \(\lambda I)\mathbf{v} = 0\). For the first eigenvalue \(\lambda_1\): $$ (C/G - \lambda_1 I)\mathbf{v}_1 = 0 \Rightarrow \left[\begin{array}{cc} -1-\lambda_1 & -5 \\ 20 & -1-\lambda_1 \end{array}\right] \mathbf{v}_1 = 0 $$ After substituting \(\lambda_1\), we can write the augmented matrix for this system and solve it by Gaussian elimination: $$ \left[\begin{array}{cc|c} -1-\frac{-2 + \sqrt{408}}{2} & -5 & 0 \\ 20 & -1-\frac{-2 + \sqrt{408}}{2} & 0 \end{array}\right] $$ Then, we can normalize the eigenvector we find. Let's denote it by \(\mathbf{U}\). The same process must be followed for the second eigenvalue \(\lambda_2\): $$ (C/G - \lambda_2 I)\mathbf{v}_2 = 0 \Rightarrow \left[\begin{array}{cc} -1-\lambda_2 & -5 \\ 20 & -1-\lambda_2 \end{array}\right] \mathbf{v}_2 = 0 $$ After substituting \(\lambda_2\), we can write the augmented matrix for this system and solve it by Gaussian elimination: $$ \left[\begin{array}{cc|c} -1-\frac{-2 - \sqrt{408}}{2} & -5 & 0 \\ 20 & -1-\frac{-2 - \sqrt{408}}{2} & 0 \end{array}\right] $$ Normalize the eigenvector we find and denote it by \(\mathbf{V}\).

4. Plot a typical shadow trajectory and the resulting axes of symmetry

Using a software or graphing tool plot the shadow trajectory for the given linear system, and add the vectors \(\mathbf{U}\) and \(\mathbf{V}\). The shadow trajectories will show the behavior of the system and how the axes of symmetry affect the trajectories. By completing these steps, you can find the vectors \(\mathbf{U}\) and \(\mathbf{V}\) parallel to the axes of symmetry of the shadow trajectories and plot a typical trajectory.

Key concepts

Characteristic Equation

The characteristic equation is a key concept when studying eigenvalues and eigenvectors. In the given exercise, you are finding the eigenvalues of a matrix, which involves solving the characteristic equation. This equation is derived from the matrix by taking the determinant of \(C/G - \lambda I\), where \(\lambda\) represents the eigenvalues and \(I\) is the identity matrix. By setting the determinant to zero, we obtain an equation called the characteristic equation: \(\text{det}(C/G - \lambda I) = 0\). Solving this equation gives us the eigenvalues of the matrix, which reflect the scaling factors that describe how the eigenvectors of the matrix stretch and compress.

Quadratic Formula

Once the characteristic equation is established, solving it often requires the use of the quadratic formula, especially when it is a polynomial of degree two. The quadratic formula is given by:

• \(\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In our example, the characteristic equation results in a quadratic form \(\lambda^2 + 2\lambda - 101 = 0\). Here, \(a = 1\), \(b = 2\), and \(c = -101\). Substituting these into the quadratic formula helps find the eigenvalues. The quadratic formula not only helps find complex eigenvalues but ensures all possible solutions for \(\lambda\) are covered.

Gaussian Elimination

Finding eigenvectors involves solving a system of linear equations, which is often done using Gaussian elimination. This method systematically performs row operations to bring the augmented matrix to a simpler form, often row echelon form or reduced row echelon form. In the exercise, after calculating each eigenvalue \(\lambda_1\) and \(\lambda_2\), the augmented matrices are set up for each and Gaussian elimination is applied. This process is essential to isolate variables and solve for the eigenvectors that correspond to each eigenvalue. The resulting vectors \(\mathbf{U}\) and \(\mathbf{V}\) represent directions that reflect the axes of symmetry of the given system.

Linear System Analysis

Understanding eigenvalues and eigenvectors is crucial for analyzing linear systems. In this problem, the linear matrix system is analyzed through these lenses to determine the behavior of trajectories. Eigenvalues give insight into the system's stability and how solutions evolve over time. Eigenvectors provide directions associated with these growth rates. By computing \(\mathbf{U}\) and \(\mathbf{V}\), you identify the axes of symmetry, which show how the shadow trajectories align with the system's dynamics. Plotting these trajectories offers a visual representation that aids in understanding how the system behaves and reacts to changes.