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Characteristic Polynomial — Definition, Formula & Examples

The characteristic polynomial of a square matrix is a polynomial whose roots are the eigenvalues of that matrix. You find it by computing the determinant of the matrix A−λIA - \lambda I, where λ\lambda is a variable and II is the identity matrix.

Given an n×nn \times n matrix AA, the characteristic polynomial is defined as p(λ)=det⁡(A−λI)p(\lambda) = \det(A - \lambda I), where II is the n×nn \times n identity matrix. This polynomial has degree nn in λ\lambda, and its roots (real or complex) are precisely the eigenvalues of AA.

Key Formula

p(λ)=det⁡(A−λI)p(\lambda) = \det(A - \lambda I)
Where:
  • AA = An n × n square matrix
  • λ\lambda = A scalar variable (eigenvalue parameter)
  • II = The n × n identity matrix

How It Works

To find the characteristic polynomial, subtract λ\lambda from every diagonal entry of AA, then compute the determinant of the resulting matrix. The determinant produces a polynomial expression in λ\lambda. Setting this polynomial equal to zero gives the characteristic equation, and solving it yields the eigenvalues. For a 2×22 \times 2 matrix, the result is always a quadratic; for a 3×33 \times 3 matrix, a cubic; and so on.

Worked Example

Problem: Find the characteristic polynomial of A = [[4, 1], [2, 3]].
Step 1: Form the matrix A−λIA - \lambda I by subtracting λ\lambda from each diagonal entry.
A−λI=[4−λ123−λ]A - \lambda I = \begin{bmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{bmatrix}
Step 2: Compute the determinant of this matrix.
det⁡(A−λI)=(4−λ)(3−λ)−(1)(2)=12−7λ+λ2−2\det(A - \lambda I) = (4 - \lambda)(3 - \lambda) - (1)(2) = 12 - 7\lambda + \lambda^2 - 2
Step 3: Simplify the expression.
p(λ)=λ2−7λ+10=(λ−5)(λ−2)p(\lambda) = \lambda^2 - 7\lambda + 10 = (\lambda - 5)(\lambda - 2)
Answer: The characteristic polynomial is λ2−7λ+10\lambda^2 - 7\lambda + 10, which factors as (λ−5)(λ−2)(\lambda - 5)(\lambda - 2). The eigenvalues are λ=5\lambda = 5 and λ=2\lambda = 2.

Why It Matters

The characteristic polynomial is the standard method for finding eigenvalues, which appear throughout engineering, physics, and data science. In structural analysis, eigenvalues determine natural frequencies of vibration; in machine learning, principal component analysis relies on eigenvalues to reduce dimensionality.

Common Mistakes

Mistake: Computing det⁡(λI−A)\det(\lambda I - A) instead of det⁡(A−λI)\det(A - \lambda I) and worrying about the sign difference.
Correction: Both conventions are used in textbooks. They differ only by a factor of (−1)n(-1)^n. The roots (eigenvalues) are identical, so follow whichever convention your course uses and be consistent.