Linear Algebra · Proof

Eigenvalues and Eigenvectors: Intuition and Computation

Q: What is an eigenvalue?

An eigenvalue λ is the scalar factor by which the corresponding eigenvector is scaled. If λ = 2, the eigenvector doubles in length. If λ = −1, it is reflected. If λ = 0, it maps to the zero vector (meaning A is singular).

Q: How are eigenvalues found?

From Av = λv we get (A − λI)v = 0. For a non-zero solution v to exist, the matrix A − λI must be singular, i.e. det(A − λI) = 0. This characteristic equation is a polynomial in λ whose roots are the eigenvalues.

Q: Can a matrix have complex eigenvalues?

Yes. A real matrix can have complex eigenvalues, which always come in conjugate pairs. For 2×2 matrices, the characteristic equation is quadratic; its discriminant determines whether eigenvalues are real or complex.

Most vectors get both rotated and scaled by a matrix transformation. Eigenvectors are the special directions that are only scaled — the transformation leaves their line of action invariant. Eigenvalues tell you by how much.

Geometric Intuition

When a matrix A acts on a typical vector v, the output Av points in a completely different direction. But for certain special vectors, the output is simply a scaled copy of the input:

A·v = λ·v (v ≠ 0)

Such a vector v is called an eigenvector of A, and the scalar λ is the corresponding eigenvalue. Geometrically, the eigenvector defines a line through the origin that A maps to itself.

Examples of what different eigenvalues mean:

λ = 3: the eigenvector is stretched to three times its length.
λ = 1: the eigenvector is unchanged.
λ = −1: the eigenvector is reflected through the origin.
λ = 0: the eigenvector is collapsed to zero (A is singular).

The Characteristic Equation

Starting from A·v = λ·v, rearrange:

A·v − λ·v = 0 (A − λI)·v = 0

For a non-zero eigenvector v to satisfy (A − λI)v = 0, the matrix (A − λI) must fail to be invertible — otherwise the only solution would be v = 0. A square matrix is non-invertible exactly when its determinant is zero:

det(A − λI) = 0

This is the characteristic equation. For a 2×2 matrix A = [[a, b], [c, d]], it expands to:

det([[a−λ, b], [c, d−λ]]) = (a−λ)(d−λ) − bc = 0 λ² − (a+d)λ + (ad−bc) = 0

The sum a+d is the trace of A and ad−bc is the determinant of A. The two roots of this quadratic are the eigenvalues.

Worked Example: 2×2 Matrix

Find the eigenvalues and eigenvectors of A = [[3, 1], [0, 2]].

Step 1: Characteristic equation.

det([[3−λ, 1], [0, 2−λ]]) = (3−λ)(2−λ) − 0 = λ² − 5λ + 6 = 0

Factoring: (λ−3)(λ−2) = 0, so λ₁ = 3 and λ₂ = 2.

Step 2: Eigenvector for λ₁ = 3.

(A − 3I)·v = 0 → [[0, 1], [0, −1]]·v = 0

The second row gives 0·v₁ − 1·v₂ = 0, so v₂ = 0. Choose v₁ = 1:

v₁ = (1, 0)

Step 3: Eigenvector for λ₂ = 2.

(A − 2I)·v = 0 → [[1, 1], [0, 0]]·v = 0

The first row gives v₁ + v₂ = 0, so v₁ = −v₂. Choose v₂ = 1:

v₂ = (−1, 1)

Verification:

A·(1,0) = (3,0) = 3·(1,0). ✓
A·(−1,1) = (−3+1, 0+2) = (−2,2) = 2·(−1,1). ✓

Frequently Asked Questions

What is an eigenvector?

An eigenvector of a matrix A is a non-zero vector v such that Av = λv for some scalar λ. The transformation A merely scales the vector v by λ — it does not rotate or change its direction (though it may flip it if λ < 0).

What is an eigenvalue?

An eigenvalue λ is the scalar factor by which the corresponding eigenvector is scaled. If λ = 2, the eigenvector doubles in length. If λ = −1, it is reflected. If λ = 0, it maps to the zero vector (meaning A is singular).

How are eigenvalues found?