Eigenvalues & Eigenvectors
“An eigenvector is the direction that doesn't wobble under a map; the eigenvalue is how much it scales.”
The formula
Av = λv (v ≠ 0); det(A − λI) = 0How to read it: an eigenvector v keeps its direction under A and is only scaled by λ; that factor λ is the eigenvalue
- A
- — a square matrix—a transformation of space
- v
- — the eigenvector—a vector whose direction is unchanged by the map
- λ
- — the eigenvalue—the factor by which that direction stretches or shrinks
- det(A−λI)=0
- — the characteristic equation that finds the eigenvalues
The hook
However a matrix spins space around, there are special axes that never change direction.
In plain words
Apply the transformation A and most vectors get knocked off their direction. But certain special vectors v keep their direction and only get scaled by a factor λ. Such a v is an eigenvector, and the factor λ is its eigenvalue.
The intuition
See a transformation as 'stretching and turning space', and eigenvectors are like its axis of rotation—the directions that do not wobble. Just as points on Earth's spin axis keep their direction as the planet turns, an eigenvector's direction is preserved by the map. λ says how much that axis stretches.
How it's built
Rearranging Av=λv gives (A−λI)v=0. For this to have a solution with v≠0, the matrix A−λI must be 'flat', and that condition is det(A−λI)=0. Solving this characteristic equation for λ gives the eigenvalues; plugging each λ back gives the eigenvectors.
Example
For A=[[2,1],[1,2]]: det([[2−λ,1],[1,2−λ]]) = (2−λ)²−1 = λ²−4λ+3 = (λ−1)(λ−3) = 0, so λ=1, 3. The eigenvector for λ=3 is (1,1): A(1,1)=(3,3)=3(1,1). For λ=1 it is (1,−1): A(1,−1)=(1,−1).
Common mistake
An eigenvector is not an arrow of a particular length—it is a direction. If (1,1) is an eigenvector, so are (2,2) and (−5,−5): all the same eigenvector (same direction). What matters is the direction and its scaling λ, not the size.
Where it's used
Google PageRank (an eigenvector of the web), natural frequencies of vibration and resonance, energy levels in quantum mechanics, principal component analysis (PCA), structural stability of bridges and buildings—eigenvalues appear whenever you seek a system's 'natural axes'.
Where it came from
Euler planted the seed studying rotation, and Hilbert attached 'eigen' (German for 'own') in the early 1900s. With quantum mechanics it became a core language of physics.
Prerequisites
Quick check
Which is one eigenvalue of the diagonal matrix [[3,0],[0,5]]?
- 4
- 3✓
- 0
- 8
Practice
Find the larger of the two eigenvalues of A=[[2,0],[0,7]].
Answer: 7
- the eigenvalues of a diagonal matrix are its diagonal entries 2 and 7
- the larger is 7
Takeaway: For diagonal or triangular matrices the diagonal entries are the eigenvalues.
Find the eigenvalue of the upper-triangular matrix A=[[4,1],[0,4]].
Answer: 4
- triangular matrix eigenvalues are the diagonal entries
- both are 4 (a repeated eigenvalue)
Takeaway: For a triangular matrix det(A−λI) factors straight into the diagonal entries.
Solve det(A−λI)=0 for A=[[2,1],[1,2]] and find the larger eigenvalue.
Answer: 3
- (2−λ)² − 1 = λ² − 4λ + 3
- = (λ−1)(λ−3) = 0 → λ=1, 3
- the larger is 3
Takeaway: The characteristic equation det(A−λI)=0 yields the eigenvalues.
Verify that (1,1) is an eigenvector of A=[[2,1],[1,2]] and find its eigenvalue.
Answer: undefined
- A(1,1) = (2·1+1·1, 1·1+2·1) = (3,3)
- (3,3) = 3·(1,1), so direction is preserved, eigenvalue λ=3
Takeaway: To check an eigenvector, just see whether Av is a scalar multiple of v.