How this calculator works
The characteristic polynomial is det(λI - A), a polynomial in λ whose degree equals the matrix size. Its roots are the eigenvalues. The two common sign conventions differ by a factor of -1 when the size is odd. This calculator builds the coefficients with the Faddeev-LeVerrier recurrence, which uses only matrix multiplication and traces. The constant term is the determinant up to a sign, and the coefficient of λ^(n-1) is the negative trace.
Formula
p(λ) = det(λI - A) = λ^n - tr(A) λ^(n-1) + ... + (-1)^n det(A). For 2 by 2, p(λ) = λ^2 - tr(A)λ + det(A).
Worked example
For the matrix with rows 1, 2 and 3, 4, the trace is 5 and the determinant is -2, so p(λ) = λ^2 - 5λ - 2. For the diagonal 3 by 3 matrix with diagonal 2, 3, 4, p(λ) = (λ - 2)(λ - 3)(λ - 4) = λ^3 - 9λ^2 + 26λ - 24.
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Frequently asked questions
Why are there two definitions?
Some texts use det(λI - A) and others use det(A - λI). The roots are the same, and the polynomials differ only by -1 when the matrix size is odd.
What do the roots tell me?
Each root is an eigenvalue with at least one eigenvector. Repeated roots count with algebraic multiplicity, and a repeated root does not always give enough eigenvectors for diagonalization.
Can the coefficients be irrational?
Yes, for matrices with decimal entries or with eigenvalues that are not integers. The coefficients are displayed as floating-point numbers.