How to use this calculator
- Enter or choose matrix A.
- The eigenvalues λ updates as you type. Use Copy result or Share to keep it.
Method
p(λ) = det(λI − A) by the Faddeev–LeVerrier recursion: Mₖ = A·Mₖ₋₁ + cₙ₋ₖ₊₁·I, cₙ₋ₖ = −tr(A·Mₖ) ÷ k; eigenvalues are the roots of p(λ)
- A
- the n × n matrix
- I
- the identity matrix
- cₖ
- coefficients of the characteristic polynomial
- λ
- an eigenvalue
Worked example
- Trace = 4 + 3 + 2 = 9 and det A = 4(3·2 − 1·0) − 1(0·2 − 1·1) + 2(0·0 − 3·1) = 24 + 1 − 6 = 19.
- The sum of the 2 × 2 principal minors is (12 − 0) + (8 − 2) + (6 − 0) = 24, so p(λ) = λ³ − 9λ² + 24λ − 19.
- Its roots are λ ≈ 1.4679, 2.6527 and 4.8794; they add to 9 and multiply to 19.
The matrix has three real eigenvalues, about 1.468, 2.653 and 4.879.
What the result means
Eigenvalues tell you how the matrix stretches space along its eigenvectors. Complex eigenvalues of a real matrix come in conjugate pairs and indicate rotation. The spectral radius decides whether repeated multiplication by A grows (> 1) or dies out (< 1).
Assumptions and limits
- Entries are real numbers, read exactly as decimals or fractions.
- Up to 10 × 10.
- Eigenvectors are not computed.
- Eigenvalues are numerical (10 significant figures); the characteristic polynomial is exact.
Questions
What does multiplicity mean here?
It is the algebraic multiplicity: how many times λ is a root of the characteristic polynomial. It is computed exactly, so a repeated eigenvalue is not reported as two nearly equal numbers.
Why are the eigenvalues of a symmetric matrix real?
The spectral theorem: a real symmetric matrix can be diagonalised by an orthogonal matrix, so all its eigenvalues are real.