What Are Eigenvalues and Eigenvectors?

Imagine you have a transformation that stretches or rotates space. An eigenvector is a special vector that only gets stretched (not rotated), and the eigenvalue tells you how much it's stretched.

Mathematically: \[ A\mathbf{v} = \lambda \mathbf{v} \] where \( \mathbf{v} \) is the eigenvector and \( \lambda \) is the eigenvalue.

Why Are They Useful?

• In physics, they describe vibrations and stability.
• In computer science, they help with search engines and face recognition.
• In economics, they model systems that evolve over time.

How to Find Them

• Solve \( (A - \lambda I)\mathbf{v} = 0 \).
• This usually means solving a polynomial equation for \( \lambda \) (the eigenvalue).

Real-Life Connections

• Google's search algorithm uses eigenvectors!
• Engineers use them to analyze buildings and bridges for safety.