The characteristic polynomial encodes the eigenvalues of a matrix as its roots, and by the Cayley–Hamilton theorem every matrix satisfies its own characteristic polynomial. Its coefficients include the trace and the determinant.
How to use the Characteristic Polynomial Calculator
- Choose the matrix size – the matching grid of fields appears.
- Enter the entries row by row (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The coefficients are computed with the Faddeev–LeVerrier algorithm and the roots found numerically. The two common conventions differ only by the sign (−1)ⁿ.
Formula
p(λ) = det(A − λI)
2 × 2: λ² − tr(A)·λ + det(A)
Worked example
Inputs — Matrix size: 3 × 3; a₁: 4; b₁: 1; c₁: 2; a₂: 1; b₂: 3; c₂: 0; a₃: 2; b₃: 0; c₃: 5; Definition: det(A − λI).
Results — Characteristic polynomial = det(A − λI) = −λ³ + 12λ² − 42λ + 43; Coefficients (of det(λI − A), highest power first) = 1, −12, 42, −43; Roots (eigenvalues) = 1.8548973, 3.4760236, 6.6690791.
Key terms
- Characteristic polynomial
- det(λI − A), whose roots are the eigenvalues.
Frequently asked questions
What are the roots of the characteristic polynomial?
The eigenvalues of the matrix.
Which convention is correct?
Both are used; det(λI − A) is monic, det(A − λI) differs by (−1)ⁿ.
What is the Cayley–Hamilton theorem?
Substituting A into its characteristic polynomial gives the zero matrix.
How are the trace and determinant related to it?
For 2 × 2: λ² − tr(A)λ + det(A).