Eigenvalues and eigenvectors are fundamental concepts in linear algebra, particularly useful in the context of 3D mathematics and computer graphics. They provide insights into the properties of linear transformations and are widely used in various applications, including 3D graphics, stability analysis, and more.

• Eigenvalue (λ): A scalar that indicates how much the eigenvector is stretched or compressed during the linear transformation.
• Eigenvector (v): A non-zero vector that only changes by a scalar factor when a linear transformation is applied.

The relationship between a matrix \(A\), an eigenvector \(v\), and its corresponding eigenvalue \(λ\) is given by: \[ A \mathbf{v} = \lambda \mathbf{v} \]

To find the eigenvalues of a matrix \(A\), we solve the characteristic equation: \[ \det(A - \lambda I) = 0 \] where \(I\) is the identity matrix of the same dimension as \(A\).

• Eigenvectors corresponding to distinct eigenvalues are linearly independent.
• The sum of the eigenvalues of a matrix equals the trace of the matrix.
• The product of the eigenvalues of a matrix equals the determinant of the matrix.

Let's consider a 2x2 matrix \(A\) and find its eigenvalues and eigenvectors.

\[ A = \begin{pmatrix} 4 & 1 \ 2 & 3 \end{pmatrix} \]

Step 1: Find the Characteristic Equation

\[ \det(A - \lambda I) = 0 \] \[ \det \begin{pmatrix} 4 - \lambda & 1 \ 2 & 3 - \lambda \end{pmatrix} = 0 \] \[ (4 - \lambda)(3 - \lambda) - (2 \cdot 1) = 0 \] \[ \lambda^2 - 7\lambda + 10 = 0 \]

Step 2: Solve for Eigenvalues

\[ \lambda^2 - 7\lambda + 10 = 0 \] \[ (\lambda - 5)(\lambda - 2) = 0 \] So, the eigenvalues are: \[ \lambda_1 = 5 \] \[ \lambda_2 = 2 \]

Step 3: Find Eigenvectors

For each eigenvalue, solve \( (A - \lambda I) \mathbf{v} = 0 \).

Eigenvector for \( \lambda_1 = 5 \)

\[ (A - 5I) \mathbf{v} = 0 \] \[ \begin{pmatrix} 4 - 5 & 1 \ 2 & 3 - 5 \end{pmatrix} \begin{pmatrix} v_1 \ v_2 \end{pmatrix} = 0 \] \[ \begin{pmatrix} -1 & 1 \ 2 & -2 \end{pmatrix} \begin{pmatrix} v_1 \ v_2 \end{pmatrix} = 0 \] \[ -v_1 + v_2 = 0 \] \[ v_2 = v_1 \] So, an eigenvector corresponding to \( \lambda_1 = 5 \) is: \[ \mathbf{v}_1 = \begin{pmatrix} 1 \ 1 \end{pmatrix} \]

Eigenvector for \( \lambda_2 = 2 \)

\[ (A - 2I) \mathbf{v} = 0 \] \[ \begin{pmatrix} 4 - 2 & 1 \ 2 & 3 - 2 \end{pmatrix} \begin{pmatrix} v_1 \ v_2 \end{pmatrix} = 0 \] \[ \begin{pmatrix} 2 & 1 \ 2 & 1 \end{pmatrix} \begin{pmatrix} v_1 \ v_2 \end{pmatrix} = 0 \] \[ 2v_1 + v_2 = 0 \] \[ v_2 = -2v_1 \] So, an eigenvector corresponding to \( \lambda_2 = 2 \) is: \[ \mathbf{v}_2 = \begin{pmatrix} 1 \ -2 \end{pmatrix} \]

Find the eigenvalues and eigenvectors of the following matrix: \[ B = \begin{pmatrix} 3 & 2 \ 2 & 3 \end{pmatrix} \]

Step 1: Characteristic Equation

\[ \det(B - \lambda I) = 0 \] \[ \det \begin{pmatrix} 3 - \lambda & 2 \ 2 & 3 - \lambda \end{pmatrix} = 0 \] \[ (3 - \lambda)^2 - 4 = 0 \] \[ \lambda^2 - 6\lambda + 5 = 0 \] \[ (\lambda - 5)(\lambda - 1) = 0 \] Eigenvalues: \[ \lambda_1 = 5 \] \[ \lambda_2 = 1 \]

Step 2: Eigenvectors

For \( \lambda_1 = 5 \)

\[ (B - 5I) \mathbf{v} = 0 \] \[ \begin{pmatrix} 3 - 5 & 2 \ 2 & 3 - 5 \end{pmatrix} \begin{pmatrix} v_1 \ v_2 \end{pmatrix} = 0 \] \[ \begin{pmatrix} -2 & 2 \ 2 & -2 \end{pmatrix} \begin{pmatrix} v_1 \ v_2 \end{pmatrix} = 0 \] \[ -2v_1 + 2v_2 = 0 \] \[ v_2 = v_1 \] Eigenvector: \[ \mathbf{v}_1 = \begin{pmatrix} 1 \ 1 \end{pmatrix} \]

For \( \lambda_2 = 1 \)

\[ (B - I) \mathbf{v} = 0 \] \[ \begin{pmatrix} 3 - 1 & 2 \ 2 & 3 - 1 \end{pmatrix} \begin{pmatrix} v_1 \ v_2 \end{pmatrix} = 0 \] \[ \begin{pmatrix} 2 & 2 \ 2 & 2 \end{pmatrix} \begin{pmatrix} v_1 \ v_2 \end{pmatrix} = 0 \] \[ 2v_1 + 2v_2 = 0 \] \[ v_2 = -v_1 \] Eigenvector: \[ \mathbf{v}_2 = \begin{pmatrix} 1 \ -1 \end{pmatrix} \]

In this section, we explored the concepts of eigenvalues and eigenvectors, learned how to derive the characteristic equation, and solved for eigenvalues and eigenvectors with practical examples. Understanding these concepts is crucial for manipulating and interpreting linear transformations in 3D space, which is foundational for advanced topics in 3D graphics and simulations.