Eigenvalue and Eigenvector Calculator
Enter the four entries of the matrix [a b; c d].
This result is an estimate only. Double-check important figures before relying on them.
How it is calculated
- Characteristic equation: λ² − (a + d)λ + (ad − bc) = 0
- Eigenvector for λ: (b, λ − a), or (λ − d, c) when b = 0
- Vectors are shown with length 1
For the determinant and inverse, use the matrix determinant calculator.
Worked example
Matrix [4 1; 2 3]:
- Trace 7, determinant 10
- Eigenvalues 2 and 5
- Eigenvectors (0.447, −0.894) and (0.707, 0.707)
Why your number may differ
Eigenvectors can be scaled or flipped and still be valid; only the direction matters. Only 2×2 matrices are covered.
Frequently asked questions
How are the eigenvalues found?
They solve λ² − (trace)λ + (determinant) = 0, which is the characteristic equation of a 2×2 matrix.
What is an eigenvector?
A direction the matrix only stretches or flips, by the factor λ, without turning it. A v = λ v.
What do complex eigenvalues mean?
The matrix rotates vectors, so no real direction is left unchanged, as with a pure rotation.