Eigenvalue and Eigenvector Calculator

What are eigenvalues and eigenvectors?

If a matrix A only stretches or shrinks a vector v without changing its direction, v is an eigenvector of that matrix: A · v = λ · v. The number λ is the eigenvalue. Eigenvalues decide vibration frequencies, stability, principal stresses, the steady state of a Markov chain and principal component analysis.

How are they found?

First set up the characteristic equation det(A − λI) = 0. It is a polynomial whose degree equals the size of the matrix, and its roots are the eigenvalues. For a 2×2 matrix it reads λ² − tr(A) · λ + det(A) = 0. Then, for each eigenvalue, solve (A − λI) · v = 0 by Gaussian elimination. The non-zero solutions are the eigenvectors.

Example

For A = [3 0, 8 −1] the trace is 2 and the determinant is −3. The equation is λ² − 2λ − 3 = 0, with roots λ = 3 and λ = −1. For λ = 3, A − 3I = [0 0, 8 −4], and 8x − 4y = 0 gives v = (1, 2). Check: A · (1, 2) = (3, 6) = 3 · (1, 2).

Repeated and complex eigenvalues

An eigenvalue can be a repeated root of the characteristic polynomial. It may then have fewer independent eigenvectors than its multiplicity, and the matrix cannot be diagonalized: the matrix [1 1, 0 1] has the single eigenvalue 1 and only one independent eigenvector, (1, 0). A real matrix can also have complex eigenvalues. The 90 degree rotation matrix has the eigenvalues i and −i, because a rotation keeps the direction of no real vector.

What the tool gives you

The characteristic polynomial, the eigenvalues (exact when rational, otherwise in surd or decimal form), the algebraic and geometric multiplicity of each, the eigenvectors, the residual A · v − λ · v for every eigenvector and the trace and determinant checks. When all eigenvalues are rational and the matrix is diagonalizable, the matrices P and D are written out too.

Limits

The largest size is 6×6. Cells take numbers only, so symbolic matrices are not supported. Eigenvectors of eigenvalues that are not rational are given as decimals. Generalized eigenvectors and the Jordan form are outside the scope of this tool.