Eigenvectors — Definition, Calculation, Examples

TL;DR

An eigenvector of a square matrix AAA is a nonzero vector vvv that gets stretched (not rotated) by the transformation AAA — its direction is preserved, only its length is scaled. The scaling factor λ\lambdaλ is the eigenvalue. This article covers the definition, the two-step calculation, three worked examples, and the appearance of eigenvectors in PageRank, PCA, and quantum mechanics.

A Vector That a Matrix Cannot Turn

Most vectors, when multiplied by a square matrix, change both direction and length. A small set — the eigenvectors — change only their length. Their direction is fixed by the matrix. They are the invariant directions of the transformation.

Finding the eigenvectors of a matrix is finding its hidden coordinate system — the directions along which the matrix's action becomes simple multiplication. Once you know these directions, problems involving the matrix's repeated action (powers of AAA, exponentials of AAA, differential equations) become straightforward.

What an Eigenvector Is

Given a square matrix AAA of size n×n, a nonzero vector vvv is an eigenvector of AAA if there exists a scalar λ\lambda such that

Av=λv.

The scalar λ\lambda is the eigenvalue associated with vvv. The equation says: applying AAA to vvv produces the same vector vvv, scaled by λ\lambda.

The zero vector trivially satisfies this for any λ\lambda, so by convention the zero vector is not called an eigenvector.

Quick Facts:

• Defining equation: Av=λv for a nonzero vvv.
• Eigenvalue: the scalar λ\lambda. Can be zero, negative, or complex.
• Eigenvector: the direction. Any nonzero scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.
• Characteristic equation: det⁡(A−λI)=0. Its roots are the eigenvalues.
• Number of eigenvalues: an n×n matrix has n eigenvalues counted with multiplicity.
• Grade introduced: undergraduate linear algebra.

How to Find Eigenvectors

The procedure has two stages.

Stage 1 — Find the eigenvalues

Solve the characteristic equation: det⁡(A−λI)=0.

Stage 2 — Find the eigenvector for each eigenvalue

For each λ, solve the homogeneous linear system: (A−λI)v=0.

The non-trivial solutions form the eigenspace for λ. Any nonzero element of the eigenspace is an eigenvector.

Examples of Eigenvectors

Quick. Find an eigenvector of A=(3002)

Characteristic equation: det⁡(3−λ002−λ)=(3−λ)(2−λ)=0. Eigenvalues λ=3,2.

For λ=3: (A−3I)v=(000−1)v=0 gives v=(10).

For λ=2: v=(01).

Final answer: eigenpairs (λ1,v1)=(3,(1,0)) and (λ2,v2)=(2,(0,1)).

Standard (Wrong Path First — Where Intuition Breaks). Find the eigenvectors of A=(4123)

The wrong path. The rusher computes the trace and the determinant and assumes the eigenvalues are 4 and 3. The flaw: diagonal entries are eigenvalues only for diagonal matrices.

The rescue. Compute det⁡(A−λI)=(4−λ)(3−λ)−2=λ2−7λ+10=(λ−5)(λ−2).

Eigenvalues: λ=5,2.

For λ=5, v=(1,1).

For λ=2, v=(1,−2).

Final answer:(λ,v)=(5,(1,1)) and (2,(1,−2)).

Stretch. Find the eigenvalues of the rotation matrix R=(0−110).

Characteristic equation: det⁡(−λ−11−λ)=λ2+1=0.

So λ=±i.

The complex eigenvalues are expected — a 90° rotation has no real invariant direction.

Where Eigenvectors Show Up — From Google to Quantum Mechanics

Eigenvectors are not a school exercise. They are the engine behind some of the most important algorithms and physical theories of the past century.

• Google PageRank. The principal eigenvector of the web's link matrix determines the importance of every page.
• Principal Component Analysis (PCA). Finds the directions of greatest variance in data — the eigenvectors of the covariance matrix.
• Quantum mechanics. The eigenvectors of an operator are the system's eigenstates.
• Structural engineering. Buildings' vibration frequencies correspond to the eigenvalues of the stiffness matrix.
• Image compression. SVD uses eigenvectors to compress images.
• Markov chains. The stationary distribution is the principal eigenvector of the transition matrix.

The Common Errors That Cost Marks

1. Reading diagonal entries as eigenvalues for non-diagonal matrices.

2. Returning the zero vector as an eigenvector.

3. Confusing eigenvalues and eigenvectors.

4. Missing complex eigenvalues.

The Mathematicians Who Discovered Eigenvectors

Augustin-Louis Cauchy introduced the term characteristic value. David Hilbert coined the term eigenwert. John von Neumann developed the theory of eigenvectors in infinite-dimensional spaces.

Conclusion

• An eigenvector of a square matrix is a nonzero vector that the matrix scales without rotating.
• The scaling factor is the eigenvalue λ. The defining equation is Av=λv.
• To find eigenvalues: solve det⁡(A−λI)=0; eigenvectors: solve (A−λI)v=0 for each λ.
• Common mistakes include treating diagonal entries as eigenvalues.

A Practical Next Step — Three Problems

1. Find the eigenvalues and eigenvectors of (5412).
2. Show that (1−1) is an eigenvector of (2112) and find the eigenvalue.
3. Find the eigenvalues of the rotation matrix (cos⁡θ−sin⁡θsin⁡θcos⁡θ).

Frequently Asked Questions

What is an eigenvector? A nonzero vector vvv such that Av=λv for some scalar λ.

Can an eigenvalue be zero? Yes, λ=0 means Av=0 for some nonzero v, which happens when AAA is singular.

How many eigenvectors does an n×n matrix have? Infinitely many nonzero scalar multiples of an eigenvector are also eigenvectors.

What is the characteristic equation? det⁡(A−λI)=0. Its roots are the eigenvalues of AAA.

Why are eigenvectors important? They are invariant under the matrix transformation and simplify computations.