Eigenvalues and Eigenvectors
An eigenvalue λ of a matrix A is a scalar with a non-zero vector v such that Av = λv. The calculator constructs the characteristic polynomial det(A − λI), finds its real roots, and for each eigenvalue recovers an eigenvector from the null space of A − λI.
Worked examples
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Frequently asked questions
What is the characteristic polynomial?
p(λ) = det(A − λI). Its roots are the eigenvalues of A. For an n×n matrix it has degree n.
Why might no real eigenvalues appear?
Rotations and other real matrices can have purely complex eigenvalues. This calculator reports only real roots; if the characteristic polynomial has no real roots it says so explicitly.
What if the eigenspace is multi-dimensional?
The calculator returns one representative eigenvector per eigenvalue. When an eigenvalue is repeated, the eigenspace may have more independent eigenvectors than shown.