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Characteristic Equation — Definition, Formula & Examples

The characteristic equation is the polynomial equation you solve to find the eigenvalues of a square matrix. It is formed by setting the determinant of A−λIA - \lambda I equal to zero.

For an n×nn \times n matrix AA, the characteristic equation is det⁡(A−λI)=0\det(A - \lambda I) = 0, where λ\lambda is a scalar and II is the n×nn \times n identity matrix. The left-hand side, when expanded, yields a degree-nn polynomial in λ\lambda called the characteristic polynomial, and its roots are the eigenvalues of AA.

Key Formula

det⁡(A−λI)=0\det(A - \lambda I) = 0
Where:
  • AA = An n × n square matrix
  • λ\lambda = A scalar (the eigenvalue to solve for)
  • II = The n × n identity matrix

How It Works

Start with your square matrix AA. Subtract λ\lambda times the identity matrix II from AA to get the matrix A−λIA - \lambda I. Compute the determinant of this new matrix — the result is a polynomial in λ\lambda. Set that polynomial equal to zero and solve for λ\lambda. Each solution is an eigenvalue of AA. For a 2×22 \times 2 matrix, this gives a quadratic; for a 3×33 \times 3, a cubic; and so on.

Worked Example

Problem: Find the eigenvalues of the matrix A=(4123)A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}.
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{pmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{pmatrix}
Step 2: Compute the determinant and set it equal to zero.
det⁡(A−λI)=(4−λ)(3−λ)−(1)(2)=λ2−7λ+10=0\det(A - \lambda I) = (4 - \lambda)(3 - \lambda) - (1)(2) = \lambda^2 - 7\lambda + 10 = 0
Step 3: Factor and solve the characteristic equation.
(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 AA are λ1=5\lambda_1 = 5 and λ2=2\lambda_2 = 2.

Why It Matters

Finding eigenvalues is essential in stability analysis of systems of differential equations, principal component analysis in data science, and computing matrix powers. The characteristic equation is the standard algebraic tool for extracting those eigenvalues in any linear algebra course.

Common Mistakes

Mistake: Writing det⁡(λI−A)\det(\lambda I - A) instead of det⁡(A−λI)\det(A - \lambda I) and worrying the answer is wrong.
Correction: Both forms yield the same eigenvalues. The polynomials differ at most by a factor of (−1)n(-1)^n, which does not change the roots. Either convention is valid.