📘 Chapter 9: Eigenvalues & Eigenvectors

🔍 Definition

An eigenvalue \( \lambda \) and non-zero vector \( \vec{v} \) satisfy: \[ A \vec{v} = \lambda \vec{v} \] This means \( \vec{v} \) is scaled (not rotated) by the transformation \( A \).

🧮 Finding Eigenvalues

To find eigenvalues, solve the characteristic equation: \[ \det(A - \lambda I) = 0 \] The roots \( \lambda \) are the eigenvalues.

📐 Finding Eigenvectors

Once an eigenvalue is known, plug it into: \[ (A - \lambda I)\vec{v} = 0 \] Then solve the homogeneous system for \( \vec{v} \).

📊 Example

Let \[ A = \begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix} \] Compute the characteristic polynomial: \[ \det(A - \lambda I) = (4 - \lambda)(3 - \lambda) - 2 = \lambda^2 - 7\lambda + 10 \] So the eigenvalues are \( \lambda = 5 \) and \( \lambda = 2 \). Solve each to find eigenvectors.

📝 Practice

Q: Find the eigenvalues and eigenvectors of: \[ B = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \] Hint: The eigenvalues are real and distinct.