Enter matrix values
Choose 2×2 or 3×3, enter coefficients, then compute the characteristic equation, invariants, and eigenvalue roots.
How the characteristic equation is formed
General rule: For a square matrix A, the characteristic equation is built from det(A − λI) = 0 or equivalently det(λI − A) = 0.
For a 2×2 matrix A = [[a, b], [c, d]]:
p(λ) = λ2 − (a + d)λ + (ad − bc)
For a 3×3 matrix:
p(λ) = λ3 − tr(A)λ2 + Sλ − det(A)
Here, S is the sum of the three principal 2×2 minors. The roots of p(λ) are the eigenvalues.
Steps to work with this calculator
- Select whether your matrix is 2×2 or 3×3.
- Enter each visible matrix value.
- Click Calculate Now.
- Read the characteristic equation, invariants, and eigenvalues.
- Use the graph to inspect real-axis crossings.
- Export the result as CSV or PDF when needed.
Worked sample matrices
| Matrix | Characteristic equation | Eigenvalues | Use case |
|---|---|---|---|
| [[4, 1], [2, 3]] | λ2 − 7λ + 10 = 0 | 5, 2 | Simple 2×2 practice with two real roots. |
| [[2, 1, 0], [1, 2, 0], [0, 0, 3]] | λ3 − 7λ2 + 15λ − 9 = 0 | 1, 3, 3 | Useful for repeated eigenvalue demonstrations. |
| [[0, -1], [1, 0]] | λ2 + 1 = 0 | i, -i | Shows purely complex eigenvalues. |
Common questions
1. What does the characteristic equation represent?
It is the polynomial equation obtained from det(A − λI) = 0. Its roots are the eigenvalues of the matrix, which describe important transformation behavior.
2. Why are trace and determinant shown?
They are key invariants. In 2×2 matrices, they directly create the polynomial. In 3×3 matrices, they still help build and verify the characteristic form.
3. Can this calculator handle complex eigenvalues?
Yes. If the polynomial has non-real roots, the result panel prints them in a complex-number format using real and imaginary parts.
4. What is the S coefficient in the 3×3 result?
It is the sum of the three principal 2×2 minors. This middle coefficient helps complete the cubic characteristic polynomial.
5. Why might repeated eigenvalues appear?
Some matrices have the same eigenvalue more than once. The polynomial then has repeated roots, which the calculator reports directly.
6. How should I read the Plotly graph?
Where the graph crosses the horizontal axis, the polynomial equals zero. Those real crossing points correspond to real eigenvalues.
7. Is the equation always written as det(A − λI) = 0?
Not always. Some texts use det(λI − A) = 0. Both forms are equivalent up to sign conventions and produce the same eigenvalues.
8. When is this calculator most useful?
It helps during algebra study, matrix review, exam preparation, and quick verification when checking manual characteristic polynomial steps.