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Instant Calculation
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The characteristic polynomial of a matrix A, det(λI − A), is the polynomial whose roots are exactly A's eigenvalues. For a 2×2 matrix this is a simple quadratic, but for larger matrices, computing it by hand via cofactor expansion gets unwieldy fast.
This calculator uses the Faddeev-LeVerrier algorithm — a systematic recurrence based on matrix traces — to find every coefficient of the characteristic polynomial for a matrix up to 6×6.
How it works
Enter a square matrix. The tool applies the Faddeev-LeVerrier recurrence: starting from the identity matrix, it repeatedly multiplies by A, takes a trace, and uses that to build up each polynomial coefficient in turn, from the highest power of λ down to the constant term.
- Click Calculate to see your results.
Examples
A diagonal 2×2 example
For A = [[2,0],[0,3]], the characteristic polynomial is λ² − 5λ + 6, which factors as (λ−2)(λ−3) — confirming the eigenvalues are 2 and 3.
Who should use it
- Finding a matrix's determinant and trace in one step (both are visible in the polynomial's coefficients).
- A first step toward finding eigenvalues by hand for small, well-behaved matrices.
Industry applications
- Numerical linear algebra and scientific computing
- Control theory (stability analysis via characteristic polynomial roots)
Advantages
- Works for any square matrix up to 6×6, not just special cases.
- The Faddeev-LeVerrier method is exact (no iteration or convergence concerns) for computing the polynomial itself.
Limitations
- Finding the polynomial's roots (the actual eigenvalues) for degree 3 and above generally still requires a separate numerical method.
Common mistakes to avoid
- Trying to factor a higher-degree characteristic polynomial by hand when it doesn't have simple rational roots — most matrices larger than 2×2 need numerical methods to find the actual eigenvalues.
Best practices
- Use the trace (sum of the diagonal) as a quick sanity check — it should equal the negative of the second coefficient, and the sum of all the eigenvalues once you find them.
Tips
- If you need the actual eigenvalues rather than the polynomial itself, jump straight to the Eigenvalues Calculator (General Matrix), which solves for them numerically without requiring you to factor the polynomial by hand.