How this calculator works
An eigenvector v satisfies Av = λv for some number λ, called the eigenvalue. Rearranging gives (A - λI)v = 0, so eigenvectors lie in the null space of A - λI. The possible eigenvalues are the roots of the characteristic polynomial. This calculator solves that polynomial for 2 by 2 and 3 by 3 matrices, including complex roots, then finds a basis for each null space. The number of independent eigenvectors decides whether the matrix is diagonalizable.
Formula
Av = λv, equivalently det(A - λI) = 0 for the eigenvalues. Eigenvectors span null(A - λI).
Worked example
The matrix with rows 4, 1 and 2, 3 has characteristic polynomial λ^2 - 7λ + 10, so the eigenvalues are 2 and 5. For λ = 2, the null space of A - 2I is spanned by 1, -2. For λ = 5, it is spanned by 1, 1. Both eigenvectors are independent, so the matrix is diagonalizable.
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Frequently asked questions
Can eigenvalues be complex?
Yes. A real matrix can have non-real eigenvalues, which occur in conjugate pairs. The eigenvectors then also have complex entries.
Is every matrix diagonalizable?
No. A repeated eigenvalue may have too few independent eigenvectors. For example, the matrix with rows 2, 1 and 0, 2 has only one eigenvector direction.
How are eigenvectors scaled?
Any non-zero multiple of an eigenvector is also an eigenvector. This calculator divides by the first non-zero entry so the displayed direction is consistent.