Mathwords logoReference LibraryMathwords

Eigenvalue

An eigenvalue is a special scalar λ\lambda associated with a square matrix AA, where multiplying AA by some nonzero vector vv gives the same result as multiplying λ\lambda by vv. In other words, the matrix stretches or flips the vector without changing its direction.

Given a square matrix AA of size n×nn \times n, a scalar λ\lambda is called an eigenvalue of AA if there exists a nonzero vector v\mathbf{v} such that Av=λvA\mathbf{v} = \lambda\mathbf{v}. The vector v\mathbf{v} is called the eigenvector corresponding to λ\lambda. Eigenvalues are found by solving the characteristic equation det⁡(A−λI)=0\det(A - \lambda I) = 0, where II is the identity matrix. A matrix of size n×nn \times n has at most nn eigenvalues, though some may be repeated or complex.

Key Formula

det⁡(A−λI)=0\det(A - \lambda I) = 0
Where:
  • AA = a square matrix
  • λλ = the eigenvalue (the unknown to solve for)
  • II = the identity matrix of the same size as A
  • detdet = the determinant of the resulting matrix

Worked Example

Problem: Find the eigenvalues of the matrix A=(4123)A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}.
Step 1: Set up the matrix A−λIA - \lambda I by subtracting λ\lambda from each diagonal entry.
A−λI=(4−λ123−λ)A - \lambda I = \begin{pmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{pmatrix}
Step 2: Compute the determinant of A−λIA - \lambda I.
det⁡(A−λI)=(4−λ)(3−λ)−(1)(2)=λ2−7λ+10\det(A - \lambda I) = (4 - \lambda)(3 - \lambda) - (1)(2) = \lambda^2 - 7\lambda + 10
Step 3: Set the determinant equal to zero and solve the characteristic equation.
λ2−7λ+10=0\lambda^2 - 7\lambda + 10 = 0
Step 4: Factor the quadratic to find the eigenvalues.
(λ−5)(λ−2)=0  ⟹  λ=5 or λ=2(\lambda - 5)(\lambda - 2) = 0 \implies \lambda = 5 \text{ or } \lambda = 2
Answer: The eigenvalues of the matrix are λ1=5\lambda_1 = 5 and λ2=2\lambda_2 = 2.

Why It Matters

Eigenvalues appear throughout science and engineering. In physics, they describe the natural frequencies at which a structure vibrates. In data science, principal component analysis (PCA) uses eigenvalues to determine which dimensions of a dataset carry the most information. Stability analysis in differential equations also depends on whether eigenvalues are positive, negative, or complex.

Common Mistakes

Mistake: Forgetting to subtract λ\lambda from every diagonal entry of AA.
Correction: The matrix A−λIA - \lambda I changes only the diagonal entries. Each diagonal entry aiia_{ii} becomes aii−λa_{ii} - \lambda, while all off-diagonal entries stay the same.
Mistake: Allowing the zero vector as an eigenvector.
Correction: The equation Av=λvA\mathbf{v} = \lambda\mathbf{v} is trivially satisfied when v=0\mathbf{v} = \mathbf{0}, so eigenvectors must be nonzero by definition. The interesting question is whether a nonzero solution exists.

Related Terms

  • Matrix — Eigenvalues are properties of matrices
  • Determinant — Used to solve the characteristic equation
  • Square Matrix — Only square matrices have eigenvalues