Problem 5 - Differential Equations and Linear Algebra

Question

Describe the transformation of \(\mathbb{R}^{2}\) with the given matrix as a product of reflections, stretches, and shears. $$A=\left[\begin{array}{ll}0 & 2 \\\2 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The transformation of the matrix A can be described as a product of a stretching factor 2 in the direction of the eigenvector \(v_1 = \left[\begin{array}{c}1\\1\end{array}\right]\) and a reflection in the direction of the eigenvector \(v_2 = \left[\begin{array}{c}1\\-1\end{array}\right]\) followed by a stretching factor 2.

Step 1: Find the eigenvalues and eigenvectors of matrix A

To find the eigenvalues of A, we need to solve the characteristic equation, which is determined by: \[|A-\lambda I|=0\] Where \(\lambda\) represents the eigenvalue and \(I\) is the identity matrix: \[\begin{vmatrix}0-\lambda & 2 \\ 2 & 0-\lambda\end{vmatrix}=0\] Now we will solve for \(\lambda\) by calculating the determinant: \[(0-\lambda)(0-\lambda)-2\cdot 2=0\] \[\lambda^2 - 4 = 0\] This equation has two solutions for \(\lambda\), which are the eigenvalues: \[\lambda_1=2\] \[\lambda_2=-2\] Now we will find the corresponding eigenvectors. For \(\lambda_1=2\), we have: \[A v_1=2 v_1\] \[\left[\begin{array}{ll}0 & 2 \\2 & 0\end{array}\right]\left[\begin{array}{c}x_1\\x_2\end{array}\right]=2\left[\begin{array}{c}x_1\\x_2\end{array}\right]\] Solve the system of linear equations: \[\begin{cases} 2x_2=2x_1 \\ 2x_1=2x_2 \end{cases}\] We can see that \(x_1=x_2\). Taking the simplest value, we can set \(x_1=x_2=1\), so the eigenvector \(v_1=\left[\begin{array}{c}1\\1\end{array}\right]\). Now for \(\lambda_2=-2\), we have: \[A v_2=-2 v_2\] \[\left[\begin{array}{ll}0 & 2 \\2 & 0\end{array}\right]\left[\begin{array}{c}x_1\\x_2\end{array}\right]=-2\left[\begin{array}{c}x_1\\x_2\end{array}\right]\] Solve the system of linear equations: \[\begin{cases} 2x_2=-2x_1 \\ 2x_1=-2x_2 \end{cases}\] We can see that \(x_1=-x_2\). Taking the simplest value, we can set \(x_1=1\) and \(x_2=-1\), so the eigenvector \(v_2=\left[\begin{array}{c}1\\-1\end{array}\right]\).

Step 2: Describe the transformation

As we have two eigenvalues \(\lambda_1=2\) and \(\lambda_2=-2\) and their respective eigenvectors, we can describe the transformation: 1. Stretching factor 2 in the direction of the eigenvector \(v_1\). 2. Stretching factor -2 in the direction of the eigenvector \(v_2\). This means a reflection in the direction of \(v_2\) and a stretching factor 2. Therefore, the transformation of the matrix A can be described as a product of a stretching factor 2 in the direction of \(v_1\) and a reflection in the direction of \(v_2\) followed by a stretching factor 2.

Key Concepts

Eigenvalues

Eigenvalues are fundamental to understanding matrix transformations, especially in linear algebra. They are scalars that give information about how a linear transformation acts on a vector. In simpler terms, eigenvalues measure the factor by which an eigenvector is stretched or shrunk under a transformation.
To find eigenvalues, we solve the characteristic equation,

• This equation is formed by setting the determinant of \(A-\lambda I=0\), where \(A\) is the matrix in question and \(I\) is the identity matrix.
• Eigenvalues \(\lambda\) are the solutions to this equation.

In our example, the matrix \[A=\begin{bmatrix} 0 & 2 \ 2 & 0 \end{bmatrix} \] has eigenvalues \(\lambda_1=2\) and \(\lambda_2=-2\).
These values indicate:

• \(\lambda_1 = 2 \) implies a stretch by a factor of 2
• \(\lambda_2 = -2\) implies a reflection and a stretch by a factor of 2

Understanding eigenvalues helps us predict the transformation effect without computing every detail separately.

Eigenvectors

Eigenvectors are vectors that remain parallel after a transformation by matrix \(A\). They are associated with eigenvalues, and every eigenvalue has a corresponding eigenvector.
Mathematically, if \(v\) is an eigenvector of matrix \(A\) associated with eigenvalue \(\lambda\), the equation is \[\]\[A \mathbf{v} = \lambda \mathbf{v}\]\[\] Key points to understand about eigenvectors:

• Eigenvectors point in directions that are unchanged by a transformation except for their magnitude, which is scaled by the eigenvalue.
• They essentially give the directions along which transformation stretches or shrinks the space.

In our example, we found two eigenvectors:

• \(v_1 = \begin{bmatrix} 1 \ 1 \end{bmatrix}\)
• \(v_2 = \begin{bmatrix} 1 \ -1 \end{bmatrix}\)

Eigenvectors help us visually understand where and how the matrix transformation takes place, giving precise directions of operation.

Linear Algebra

Linear Algebra is the branch of mathematics concerning linear equations, matrices, and their transformation properties. It lays the foundation for understanding concepts like eigenvalues and eigenvectors.
Here’s what Linear Algebra covers:

• It deals with vector spaces and linear mappings between them, allowing the description and solutions of systems of linear equations.
• Understanding matrices, as they are the cornerstone of linear transformations, which redefine vector directions and magnitudes systematically.
• Identifies and computes eigenvalues and eigenvectors to measure transformations.

Linear Algebra is integral to fields such as:

• Engineering, for defining physical system models
• Computer Graphics, to perform transformations in 3D rendering
• Data Science, where dimensionality reduction techniques require understanding of these concepts

In our matrix problem, Linear Algebra principles enable the breaking down of complex matrix operations into systematic entities like eigenvalues and eigenvectors, enhancing powerful computations.