The characteristic polynomial of A is det(A − λI), written out as a polynomial in λ. Press Char. polynomial(A) and open the steps to see the coefficients. Its roots are exactly the eigenvalues, which is where every hand computation of eigenvalues begins.
Where it comes from
The defining equation A·v = λ·v rearranges to (A − λI)·v = 0. A non-zero v can only solve that if A − λI is singular — otherwise the only solution is v = 0 — and a matrix is singular exactly when its determinant vanishes. So the eigenvalues are the values of λ making det(A − λI) = 0, and that expression is the characteristic polynomial.
For an n×n matrix the result has degree n, and two of its coefficients can be read off the matrix without any expansion at all: the coefficient just below the leading term is the negative of the trace, and the constant term is the determinant up to a sign of (−1)ⁿ. Both make excellent checks on a hand computation.
Worked example
| 2 | 1 |
| 1 | 2 |
| 2 − λ | 1 |
| 1 | 2 − λ |
The roots 1 and 3 are the eigenvalues. Note the checks: the trace is 4 and the λ coefficient is −4; the determinant is 3 and so is the constant term. The calculator uses the Faddeev–LeVerrier recurrence, which builds the coefficients from traces of successive powers and stays exact instead of expanding a symbolic determinant.
Common mistakes
- Subtracting λ from every entry. λI is λ on the diagonal and zero elsewhere, so only the diagonal changes.
- Losing the sign convention. det(A − λI) and det(λI − A) differ by (−1)ⁿ. Both are used; pick one and stay with it.
- Reading multiplicity as a count of eigenvectors. A double root need not bring two independent eigenvectors — when it does not, the matrix is defective.
- Expecting real roots. A real matrix can have a characteristic polynomial with complex roots, and then its eigenvalues are complex.