About the Eigenvalue & Eigenvector Calculator
This eigenvalue calculator finds the eigenvalues and eigenvectors of a square matrix up to 4×4. It first builds the characteristic polynomial det(λI − A) exactly, then solves it for every eigenvalue — real or complex — and reports each one with its algebraic multiplicity. For each eigenvalue it solves (A − λI)v = 0 to give a basis of eigenvectors, scaled to whole numbers where possible.
It is built for linear algebra students checking homework, and for engineers, physicists and data scientists who need eigenvalues for stability analysis, vibration modes, Markov chains or principal component analysis. The trace and determinant are shown as a quick check: they always equal the sum and the product of the eigenvalues.
Rational eigenvalues and their eigenvectors are computed exactly with fractions. Irrational and complex eigenvalues are found numerically and shown to about six significant decimals, which is more than enough for checking answers but may differ in the last digit from a symbolic result such as (5 + √33)/2.
How to use the eigenvalue & eigenvector calculator
- 1Type the square matrix with one row per line (2×2 up to 4×4).
- 2Read the eigenvalues and the characteristic polynomial.
- 3Read the eigenvector basis for each eigenvalue (any non-zero multiple is also an eigenvector).
- 4Check the table for multiplicities to see whether the matrix is diagonalizable.
Formula and method
An eigenvector v of A is a non-zero vector that A only stretches, by the factor λ (the eigenvalue). Rearranging Av = λv to (A − λI)v = 0 shows that a non-zero solution exists only when A − λI is singular, i.e. when det(λI − A) = 0. That determinant is a degree-n polynomial in λ, the characteristic polynomial, and its roots are the eigenvalues.
The calculator builds the characteristic polynomial exactly with the Faddeev–LeVerrier method, finds all its roots numerically (Durand–Kerner iteration) and confirms rational roots exactly. For each eigenvalue it row-reduces A − λI and reads off a basis of the null space; the number of basis vectors is the geometric multiplicity. As a check, the eigenvalues always add up to the trace and multiply to the determinant.
- λ
- Eigenvalue
- v
- Eigenvector (non-zero, any non-zero multiple also works)
- I
- Identity matrix
- tr(A)
- Trace: sum of diagonal entries = sum of eigenvalues
Worked examples
Symmetric 2×2 matrix
det(λI − A) = (λ − 2)² − 1 = λ² − 4λ + 3 = (λ − 3)(λ − 1). For λ = 3, A − 3I = [[−1, 1], [1, −1]] gives v = (1, 1); for λ = 1, A − I = [[1, 1], [1, 1]] gives v = (−1, 1).
Non-symmetric 2×2 matrix
The trace is 7 and the determinant 10, so λ² − 7λ + 10 = 0 and λ = 5 or 2. Their sum 7 matches the trace and their product 10 matches the determinant.
3×3 matrix
The first row and column are separate, so 2 is an eigenvalue. The block [[3, 4], [4, 9]] has trace 12 and determinant 11, giving λ² − 12λ + 11 = 0, so λ = 11 or 1.
Rotation matrix with complex eigenvalues
A 90° rotation leaves no real direction unchanged. Its characteristic polynomial λ² + 1 has the complex roots ±i.
Frequently asked questions
How do you find the eigenvalues of a matrix?+
Subtract λ from each diagonal entry, take the determinant to get the characteristic polynomial, set it to zero and solve for λ. For a 2×2 matrix this is λ² − (trace)λ + det = 0.
How do you find the eigenvectors once you have the eigenvalues?+
For each eigenvalue λ, solve (A − λI)v = 0 by row reduction. The free variables give the eigenvectors; any non-zero multiple of an eigenvector is also an eigenvector, so answers can look different but be equivalent.
Can eigenvalues be complex or zero?+
Yes. Real matrices can have complex eigenvalues, which always come in conjugate pairs. A zero eigenvalue means the matrix is singular (its determinant is 0) and has no inverse.
What is the difference between algebraic and geometric multiplicity?+
Algebraic multiplicity is how many times λ repeats as a root of the characteristic polynomial; geometric multiplicity is the number of independent eigenvectors. If geometric is smaller for any eigenvalue, the matrix cannot be diagonalized.
Do symmetric matrices always have real eigenvalues?+
Yes. A real symmetric matrix has only real eigenvalues and an orthogonal set of eigenvectors, which is why covariance matrices in principal component analysis have real, non-negative eigenvalues.