Eigenvalues and Eigenvectors
Vectors only scaled by a transformation.
Key idea
Vectors only scaled by a transformation.
In depth
An eigenvector v of A satisfies Av = λv: the transformation only stretches it by λ. They reveal the axes along which A acts simply and the long-term behaviour of repeated application.
Worked example
A = [2 0; 0 3]: eigenvalues 2, 3 with eigenvectors (1, 0), (0, 1).
Formula
Real-world uses
- Finding natural vibration frequencies of a bridge
- Principal directions of stress in materials
Common mistake
Counting v = 0 as an eigenvector; it is excluded.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Eigenvalues and Eigenvectors?
Vectors only scaled by a transformation. It is a topic at the expert level in Linear Algebra.
What is the formula for Eigenvalues and Eigenvectors?
The key formula is Av = λv.
Where is Eigenvalues and Eigenvectors used in real life?
Real-world examples: Finding natural vibration frequencies of a bridge; Principal directions of stress in materials.
What should I learn before Eigenvalues and Eigenvectors?
Make sure you are comfortable with Orthogonal Matrices, the previous topic in Linear Algebra.
What is a common mistake with Eigenvalues and Eigenvectors?
Counting v = 0 as an eigenvector; it is excluded.
How can I practise Eigenvalues and Eigenvectors?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Eigenvalues and Eigenvectors?
Move on to Characteristic Polynomial next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.