Eigenvalue and eigenvector calculator
Type a square 2×2 or 3×3 matrix with rows separated by semicolons. The calculator finds the characteristic polynomial, all eigenvalues (real or complex) with their multiplicities, and eigenvectors for the real ones.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Enter the matrix row by row, for example
2 0 0; 0 3 4; 0 4 9.
- The eigenvalues appear first. Below them are the characteristic polynomial, trace, determinant and for each eigenvalue its multiplicity and eigenvectors.
How it works
The trace and determinant are computed with exact fractions. The characteristic polynomial is built from them, and its roots are found in closed form: the quadratic formula for 2×2, Cardano or the trigonometric method for 3×3 (see the cubic equation solver). Whenever an eigenvalue is a rational number the tool confirms it exactly and finds the eigenvectors by exact elimination of A − λI. For irrational eigenvalues the eigenvector is computed numerically.
Worked example
A = [2 1; 1 2]: trace 4, determinant 3, so λ² − 4λ + 3 = (λ − 3)(λ − 1) = 0.
- λ = 3: (A − 3I)v = 0 gives −x + y = 0, so v = (1, 1).
- λ = 1: x + y = 0, so v = (1, −1).
Tips
- The eigenvalues add up to the trace and multiply to the determinant; use this as a check.
- For a triangular matrix the eigenvalues are the diagonal entries.
- To solve A·x = b instead, use the linear system solver.
Frequently asked questions
What are eigenvalues and eigenvectors?
An eigenvector v of a matrix A is a non-zero vector that A only stretches, A·v = λ·v. The stretch factor λ is the eigenvalue.
How are the eigenvalues found?
They are the roots of the characteristic polynomial det(A − λI) = 0. For a 2×2 matrix that is λ² − (trace)λ + det = 0; for 3×3 it is the cubic λ³ − (trace)λ² + (sum of 2×2 principal minors)λ − det = 0, solved here like in the cubic equation solver.
Can eigenvalues be complex?
Yes, even for a real matrix. A rotation by 90°, [0 −1; 1 0], has eigenvalues i and −i. They always come in conjugate pairs.
What does algebraic multiplicity mean?
It is how many times an eigenvalue is a root of the characteristic polynomial. The number of independent eigenvectors can be smaller, as for [2 1; 0 2], which has the eigenvalue 2 twice but only the eigenvector (1, 0).
Why are the eigenvectors not unique?
Any non-zero multiple of an eigenvector is also an eigenvector. The tool shows whole-number vectors for rational eigenvalues, otherwise vectors scaled to start with 1.