Eigenvalues & Eigenvectors

Imagine you have a transformation that stretches, rotates, or shears the plane. Most vectors change both their direction and magnitude under such a transformation. But certain special vectors — called eigenvectors — only get scaled; they point in exactly the same (or exactly opposite) direction after the transformation. The factor by which they are scaled is called the eigenvalue. The word 'eigen' comes from German and means 'own' or 'characteristic,' capturing the idea that these vectors reveal something intrinsic about the matrix. Eigenanalysis — finding eigenvalues and eigenvectors — is one of the most powerful tools in all of mathematics. It lets us understand the long-run behavior of systems, compress images, analyze vibrations in engineering, and recommend videos on streaming platforms. For a 2×2 matrix, the process involves solving a quadratic called the characteristic polynomial, factoring it, and then finding associated direction vectors. Once a matrix has two distinct real eigenvalues, it can be diagonalized: written as M = PDP⁻¹ where D is a diagonal matrix of eigenvalues and P is a matrix whose columns are eigenvectors. This factorization makes it trivially easy to compute matrix powers Mⁿ = PDⁿP⁻¹, turning what would be repeated matrix multiplication into simple scalar arithmetic.

Key takeaways
  • The Defining Equation
  • The Characteristic Polynomial
  • Finding Each Eigenvector
  • Diagonalization: M = PDP⁻¹
  • Matrix Powers via Diagonalization

The Defining Equation

A non-zero vector v is an eigenvector of matrix M with eigenvalue λ if Mv = λv. This means the transformation M acts on v purely as a scaling. Rewriting: Mv − λv = 0, so (M − λI)v = 0. For a non-zero v to exist, the matrix (M − λI) must be singular, meaning its determinant is zero.

The Characteristic Polynomial

For a 2×2 matrix M = [[a, b],[c, d]], form M − λI = [[a−λ, b],[c, d−λ]]. Setting det(M − λI) = 0 gives the characteristic polynomial: (a−λ)(d−λ) − bc = 0, which expands to λ² − (a+d)λ + (ad−bc) = 0, or equivalently λ² − tr(M)λ + det(M) = 0. Solve this quadratic for the two eigenvalues λ₁ and λ₂.

Finding Each Eigenvector

For each eigenvalue λᵢ, substitute back into (M − λᵢI)v = 0 and solve for v. Because the matrix is singular by construction, the two rows of (M − λᵢI) are proportional, so you get infinitely many solutions — an entire line (eigenspace). Pick any convenient non-zero vector from this line as your eigenvector. For example, if the first row gives (a−λᵢ)v₁ + b·v₂ = 0, you can choose v₁ = b, v₂ = −(a−λᵢ), provided this isn't the zero vector.

Diagonalization: M = PDP⁻¹

If M has two distinct real eigenvalues λ₁ ≠ λ₂ with eigenvectors p₁ and p₂, construct P = [p₁ | p₂] (eigenvectors as columns) and D = diag(λ₁, λ₂). Then M = PDP⁻¹. Distinct eigenvalues guarantee the eigenvectors are linearly independent, so P is invertible. You can verify by checking MP = PD column by column: M·p₁ = λ₁p₁ and M·p₂ = λ₂p₂.

Matrix Powers via Diagonalization

Because M = PDP⁻¹, we have M² = (PDP⁻¹)(PDP⁻¹) = PD²P⁻¹ (the middle P⁻¹P cancels to I). By induction, Mⁿ = PDⁿP⁻¹. Since D is diagonal, Dⁿ = diag(λ₁ⁿ, λ₂ⁿ) — just raise each diagonal entry to the nth power. This turns an expensive computation into simple scalar exponentiation followed by two matrix multiplications.

The Key Equation

det⁡(M−λI)=λ2−tr⁡(M) λ+det⁡(M)=0\det(M - \lambda I) = \lambda^2 - \operatorname{tr}(M)\,\lambda + \det(M) = 0
  • \lambda — Eigenvalue (dimensionless): The scalar by which an eigenvector is stretched or flipped under the transformation M.
  • \operatorname{tr}(M) — Trace of M (dimensionless): The sum of the diagonal entries of M: tr(M) = a + d. It equals the sum of the eigenvalues λ₁ + λ₂.
  • \det(M) — Determinant of M (dimensionless): det(M) = ad − bc. It equals the product of the eigenvalues λ₁ · λ₂.
  • I — Identity Matrix (dimensionless): The 2×2 matrix with 1s on the diagonal and 0s elsewhere; subtracting λI shifts each diagonal entry by λ.

Worked Example

Problem. Let M = [[3, 1],[0, 2]]. Find the eigenvalues and eigenvectors of M, write the diagonalization M = PDP⁻¹, and compute M⁵.

M5=(243211032)M^5 = \begin{pmatrix} 243 & 211 \\ 0 & 32 \end{pmatrix}

Answer. M⁵ = [[243, 211],[0, 32]]

Figure
Diagram
Eigenvectors of M = [[3,1],[0,2]] under transformationxyslope = −1 linep₁=[1,0]Mp₁=3p₁=[3,0]p₂=[−1,1]Mp₂=2p₂Blue eigenvector p₁=[1,0] scaled by λ₁=3 (stays on x-axis)Red eigenvector p₂=[−1,1] scaled by λ₂=2 (stays on same line)
Origins · 1 minCarl Gustav Jacob Jacobi and how this was discovered
CG
Carl Gustav Jacob Jacobi
1804–1851 · German

Jacobi made foundational contributions to the theory of determinants and developed systematic methods for computing eigenvalues of symmetric matrices, including what became known as the Jacobi eigenvalue algorithm. He also introduced the functional determinant (Jacobian matrix), deepening the connection between linear transformations and their characteristic properties. Jacobi was so intensely focused on mathematics that he reportedly worked for days without sleep during his most productive periods. He became a full professor at Königsberg at age 23 — one of the youngest in Prussian university history.

  • Determinants of 2×2 and 3×3 Matrices — The characteristic polynomial is built from det(M − λI) = 0; you must be fluent with determinants to find eigenvalues.
  • Matrix Multiplication — Diagonalization and computing Mⁿ = PDⁿP⁻¹ both require matrix multiplication; understanding it is essential before eigenanalysis.
  • Inverse of a 2×2 Matrix — Diagonalization requires computing P⁻¹; knowing the formula for inverting a 2×2 matrix is a direct prerequisite.
  • Solving Quadratic Equations — The characteristic polynomial of a 2×2 matrix is always a quadratic in λ; students need the quadratic formula and factoring to solve it.
  • Linear Independence of Vectors — Diagonalizability requires eigenvectors to be linearly independent; distinct eigenvalues guarantee this, connecting both concepts.