Eigenvalues and Eigenvectors Calculator

Compute the eigenvalues1 and eigenvectors2 of a square matrix \(A\).
Supports real, complex, fractions (e.g. 3+2i, 1/2), and expressions like pi, sqrt(2), exp(1).

Eigenvalues and Eigenvectors:

  • For a square matrix \(A\), an eigenvalue \(\lambda\) and corresponding eigenvector \(\mathbf{v}\) satisfy \(A\mathbf{v} = \lambda\mathbf{v}\)
  • The eigenvector \(\mathbf{v}\) is a direction that gets scaled by factor \(\lambda\) when transformed by \(A\)
  • Eigenvalues are found by solving the characteristic equation3 \(\det(A - \lambda I) = 0\)
  • A matrix may have real or complex eigenvalues; complex eigenvalues of real matrices come in conjugate pairs

Generalized Eigenvectors:

  • When a matrix is defective4 (lacks enough eigenvectors to span the space), generalized eigenvectors \(\mathbf{w}\) satisfy \((A - \lambda I)^k\mathbf{w} = \mathbf{0}\) for some \(k > 1\)
  • Generalized eigenvectors complete the basis when geometric multiplicity is less than algebraic multiplicity

This calculator finds eigenvalues with algebraic multiplicities, eigenvectors scaled for clean display (surd form, fractions), and generalized eigenvectors when needed. Complex eigenvectors are displayed in \(a + bi\) form.

Set matrix dimension

Enter the matrix size n and click Generate Matrix to create an n×n input grid.

1 — Set Up
Loads a pre-filled 4×4 matrix — ready to compute.
2 — Compute
Finds all eigenvalues and corresponding eigenvectors of the square matrix.
Resets all dimensions, matrix entries, and results.