What is an eigenvalue and eigenvector?

Problem Statement

What is an eigenvalue and eigenvector?

Step-by-Step Explanation

An eigenvector keeps its direction under a matrix transformation, while the eigenvalue tells how it scales.

1. Identify what the question is asking. In Linear Algebra, this usually means naming the known quantities, the unknown quantity, and the concept that connects them.
2. Write the relevant definition, rule, or relationship before substituting numbers or manipulating symbols.
3. Work one step at a time and keep units, signs, and restrictions visible.
4. Check whether the answer makes sense in the original context.

Key Formula

Final Answer

An eigenvector keeps its direction under a matrix transformation, while the eigenvalue tells how it scales. The reliable strategy is to start from the definition, apply the correct rule carefully, and then interpret the result in plain language.