Linear Algebra
7.112 min read

What Is an Eigenvector?

Most vectors change direction when a matrix is applied to them. An eigenvector is a special nonzero vector that does not change direction — the matrix only stretches or shrinks it. The scaling factor is called the eigenvalue λ\lambda. The defining equation is Av=λvA\mathbf{v} = \lambda\mathbf{v}: apply the matrix, get back the same vector scaled by λ\lambda.

Think of a 2×22 \times 2 matrix as a machine that squishes, stretches, and rotates the plane. Most directions get twisted around. But eigenvectors are the "natural axes" of the transformation — arrows that stay on their own line. If λ=3\lambda = 3, the eigenvector gets tripled. If λ=−1\lambda = -1, it flips direction. If λ=0\lambda = 0, it maps to the zero vector.

Use the interactive diagram to explore this. Set the matrix entries and watch the gray unit vectors get transformed (blue arrows). The orange and teal eigenvectors stay along their dashed lines — they only scale. Change the matrix until the eigenvalues become complex: then there are no real eigenvectors at all, and every direction gets rotated.

Formal View

Definition 7.1 — Eigenvalue and Eigenvector
Let AA be an n×nn \times n matrix. A nonzero vector v∈Rn\mathbf{v} \in \mathbb{R}^n is an eigenvector of AA if
Av=λvA\mathbf{v} = \lambda\mathbf{v}
for some scalar λ∈R\lambda \in \mathbb{R}. The scalar λ\lambda is the eigenvalue corresponding to v\mathbf{v}.

The zero vector is excluded by definition — A0=λ0A\mathbf{0} = \lambda\mathbf{0} holds trivially for every λ\lambda, making it useless.

Example 7.2
Let A=(2003)A = \begin{pmatrix}2 & 0 \\ 0 & 3\end{pmatrix}. Then A(1,0)t=(2,0)t=2(1,0)tA(1,0)^t = (2,0)^t = 2(1,0)^t, so (1,0)t(1,0)^t is an eigenvector with λ=2\lambda = 2. Similarly, (0,1)t(0,1)^t is an eigenvector with λ=3\lambda = 3. For a diagonal matrix, the standard basis vectors are always eigenvectors, and the eigenvalues are the diagonal entries.

Interactive Visualization

Eigenvector Explorer

Why This Matters

Eigenvectors are the "natural coordinates" of a transformation — they reveal the directions along which the transformation acts most simply.

  • Face recognition (PCA/eigenfaces): the principal components of face images are eigenvectors of the covariance matrix
  • Structural engineering: the eigenvectors of a stiffness matrix give the natural vibration modes of a building
  • Quantum mechanics: every measurable physical quantity corresponds to eigenvectors of a Hermitian operator
  • Search engines: the dominant eigenvector of a web link matrix gives the PageRank importance scores

Quiz

Question 1

Which vector is an eigenvector of A=(2005)A = \begin{pmatrix}2 & 0 \\ 0 & 5\end{pmatrix}?

Question 2

The zero vector 0\mathbf{0} is an eigenvector of every matrix.

Question 3

If Av=−4vA\mathbf{v} = -4\mathbf{v} for a nonzero vector v\mathbf{v}, what is the eigenvalue?

Common Mistakes

  • Forgetting that eigenvectors must be nonzero — the zero vector satisfies A0=λ0A\mathbf{0} = \lambda\mathbf{0} for any λ\lambda and is excluded.
  • Confusing the eigenvalue λ\lambda (a scalar) with the eigenvector v\mathbf{v} (a vector).
  • Assuming eigenvectors are unique — any nonzero scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.
  • Thinking all matrices have real eigenvectors — rotation matrices, for example, have no real eigenvectors.