Enter a square matrix up to 8 × 8 to get its characteristic polynomial, its eigenvalues in exact form (fractions, square roots or complex numbers) and a basis of eigenvectors for each one, with algebraic and geometric multiplicities and a diagonalizability check.
Last updated:
Matrix A
Entries can be integers, decimals or fractions like 5/2. Empty cells count as 0.
Characteristic polynomial
–
Eigenvalues and eigenvectors
Steps
Advertisement
Advertisement
How to use the Eigenvalue and Eigenvector Calculator
Choose the size of the square matrix (2 × 2 to 8 × 8) or load an example.
Type the entries: integers, decimals or fractions such as 5/2. Empty cells count as 0.
Press Calculate eigenvalues to get the characteristic polynomial and its factored form.
Read the table: each eigenvalue with its decimal value, multiplicities and a basis of eigenvectors.
Open the steps to see A − λI and its row-reduced form for each eigenvalue, or press Copy result.
The eigenvalue equation
A non-zero vector v is an eigenvector of a square matrix A when multiplying by A only stretches it: A·v = λ·v. The scale factor λ is the eigenvalue. Rewriting gives (A − λI)v = 0, which has a non-zero solution only when A − λI is singular, so the eigenvalues are the roots of the characteristic polynomial p(λ) = det(λI − A).
2 × 2: p(λ) = λ² − (a + d)λ + (ad − bc) for A = [[a, b], [c, d]].
3 × 3: p(λ) = λ³ − tr(A)λ² + (M₁₁ + M₂₂ + M₃₃)λ − det(A), where Mᵢᵢ are the principal 2 × 2 minors.
Any size: sum of eigenvalues = trace(A), product of eigenvalues = det(A).
Worked example
For A = [[2, −1, 0], [−1, 2, −1], [0, −1, 2]] (the default matrix), trace = 6, the principal minors sum to 3 + 4 + 3 = 10 and det = 4, so p(λ) = λ³ − 6λ² + 10λ − 4 = (λ − 2)(λ² − 4λ + 2). The eigenvalues are λ = 2 and λ = 2 ± √2. Solving (A − 2I)v = 0 gives v = (1, 0, −1), and λ = 2 + √2 gives v = (1, −√2, 1). Because the matrix is symmetric, the three eigenvectors are mutually orthogonal.
Quick facts for checking your answer
Matrix type
What to expect
Triangular or diagonal
Eigenvalues are the diagonal entries
Real symmetric (A = Aᵀ)
All eigenvalues real; orthogonal eigenvectors; always diagonalizable
Rotation by θ (2 × 2)
cos θ ± i sin θ, complex unless θ is 0° or 180°
Singular (det = 0)
0 is an eigenvalue; its eigenvectors span the null space
Invertible A
A⁻¹ has eigenvalues 1/λ with the same eigenvectors
Powers Aᵏ
Eigenvalues λᵏ with the same eigenvectors
Projection (A² = A)
Eigenvalues only 0 and 1
Frequently asked questions
How do you find the eigenvalues of a matrix?
Solve the characteristic equation det(λI − A) = 0. For a 2 × 2 matrix this is λ² − (trace)λ + det = 0, so λ = (trace ± √(trace² − 4·det)) / 2. For larger matrices the characteristic polynomial has degree n and its n roots, counted with multiplicity, are the eigenvalues.
How do you find the eigenvectors once you have an eigenvalue?
Substitute the eigenvalue into A − λI and solve (A − λI)v = 0 by row reduction. The free variables of the reduced matrix give a basis of the eigenspace; any non-zero combination of those vectors is an eigenvector.
What is the difference between algebraic and geometric multiplicity?
The algebraic multiplicity is how many times λ is a root of the characteristic polynomial. The geometric multiplicity is the number of independent eigenvectors for λ (the dimension of its eigenspace). It is always between 1 and the algebraic multiplicity.
When is a matrix diagonalizable?
An n × n matrix is diagonalizable exactly when it has n independent eigenvectors, i.e. the geometric multiplicity equals the algebraic multiplicity for every eigenvalue. Distinct eigenvalues guarantee it, and real symmetric matrices are always diagonalizable with real eigenvalues.
Why do I get complex eigenvalues?
A real matrix can have a characteristic polynomial with no real roots, for example a rotation matrix. Complex eigenvalues of a real matrix always come in conjugate pairs a ± bi, and their eigenvectors are complex conjugates of each other.
Why are some eigenvalues shown as decimals?
Eigenvalues that are fractions or roots of quadratic factors are shown exactly. Roots of an irreducible cubic or higher factor have no simple closed form here, so they are computed numerically with the QR algorithm and shown to six decimals.
Is the trace really the sum of the eigenvalues?
Yes. The trace of A equals the sum of all eigenvalues and the determinant equals their product, counting multiplicity and complex values. The calculator shows both as a quick check.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.