Eigenvalue & Eigenvector Calculator
Exact eigenvalues and eigenvectors — fractions, square roots, complex pairs — with steps.
Try before you buy.
- Free preview: the size of your matrix and what the full result holds, with the eigenvalues, the working, the polynomial, eigenvectors and PDP⁻¹ hidden.
- Locked until you unlock it: download and copy.
- Unlock: Pro pass, ₹179 for 30 days, a one-time payment that never renews.
Ways to unlock shows how to get the full result.
Printing this result is locked in the free preview.
Result
This is the last result worked out. Fix the input above to update it.
Locked in the free preview. Opens the ways to unlock this result.
Locked in the free preview. Batch runs unlock with a pass.
Locked in the free preview. Query results unlock with a pass.
About the Eigenvalue & Eigenvector Calculator
Enter a square matrix up to 10 × 10 and the calculator finds its characteristic polynomial det(A − λI), the eigenvalues with their algebraic and geometric multiplicities, a basis of each eigenspace, and the diagonalisation A = PDP⁻¹ when it exists — or the Jordan block sizes when it does not.
Everything that can be exact is exact: the polynomial is computed in fractions, rational eigenvalues come out as fractions, eigenvalues from quadratic factors as square roots such as (1 + √5)/2 or complex pairs such as 1 ± 2i, and their eigenvectors are worked out in the same exact numbers. Roots of factors of degree three or more that have no closed form are found numerically, and every eigenvalue is cross-checked by the QR algorithm run on A itself. For 2 × 2 and 3 × 3 matrices the steps show the determinant expanded by hand.
How to use it
- Choose the size of the matrix and type its entries — whole numbers, decimals or fractions such as −3/4. You can paste a block copied from a spreadsheet into a cell, or switch to Text and paste rows such as [[2, 1], [1, 2]].
- Read the eigenvalues at the top. λ₂ = λ₃ = 2 means the eigenvalue 2 is a double root of the characteristic polynomial.
- Look at the table: for each eigenvalue, how many times it is a root (algebraic multiplicity), how many independent eigenvectors it has (geometric multiplicity) and a basis of its eigenvectors.
- If the matrix is diagonalisable, read P (eigenvectors as columns), D (eigenvalues on the diagonal) and P⁻¹; if not, read the Jordan form.
- Follow the steps — A − λI, the determinant, its factors and the null space of A − λI for each eigenvalue — and copy the results as text or LaTeX or download the report with a Pro pass, or after unlocking this result; without one you see the free preview.
Examples
[[4, 1], [2, 3]]
λ = 5 with eigenvector (1, 1); λ = 2 with eigenvector (1, −2)
det(A − λI) = (4 − λ)(3 − λ) − 2 = λ² − 7λ + 10 = (λ − 5)(λ − 2).
[[4, −1, 6], [2, 1, 6], [2, −1, 8]]
λ = 9 (once) and λ = 2 (twice, with two independent eigenvectors): diagonalisable
det(A − λI) = −λ³ + 13λ² − 40λ + 36 = −(λ − 9)(λ − 2)². A − 2I has rank 1, so its null space is a plane.
[[1, 1], [1, 0]]
λ = (1 ± √5)/2, the golden ratio and its conjugate
λ² − λ − 1 = 0 has discriminant 5, so the eigenvalues are irrational but exact.
[[0, −1], [1, 0]]
λ = ±i with eigenvectors (i, 1) and (−i, 1)
A real matrix with no real eigenvalue: no direction is left unturned by a rotation.
[[1, 1], [0, 1]]
λ = 1 twice but only one eigenvector (1, 0): not diagonalisable; Jordan block of size 2
Common uses
- Checking eigenvalue and diagonalisation homework, including the hand expansion of det(A − λI) for 3 × 3 matrices.
- Finding the long-run behaviour of a recurrence or a Markov chain from the eigenvalues and eigenvectors of its matrix.
- Solving systems of linear differential equations x′ = Ax, whose solutions are built from eigenvalues and eigenvectors.
- Classifying a quadratic form or a critical point from the eigenvalues of a symmetric matrix (all positive, all negative or mixed signs).
How eigenvalues and eigenvectors are found
An eigenvector of A is a non-zero vector v with Av = λv; the number λ is its eigenvalue. Rewriting this as (A − λI)v = 0 shows that A − λI must be singular, so the eigenvalues are the roots of the characteristic polynomial det(A − λI) = 0, a polynomial of degree n. For each eigenvalue, the eigenvectors are the non-zero solutions of (A − λI)v = 0 — the null space of A − λI, found by row reduction.
Two quick checks: the eigenvalues (counted with multiplicity) add up to the trace of A and multiply to its determinant. For a 2 × 2 matrix the characteristic polynomial is λ² − (tr A)λ + det A; for a triangular matrix the eigenvalues are simply its diagonal entries.
Multiplicities and diagonalisation
The algebraic multiplicity of λ is how many times it is a root of the characteristic polynomial; the geometric multiplicity is the dimension of its eigenspace (the number of independent eigenvectors). The geometric multiplicity is at least 1 and never more than the algebraic one.
A is diagonalisable — A = PDP⁻¹ with the eigenvectors as the columns of P and the eigenvalues on the diagonal of D — exactly when every geometric multiplicity equals its algebraic multiplicity, so that there are n independent eigenvectors. Distinct eigenvalues always give independent eigenvectors, so a matrix with n different eigenvalues is diagonalisable; a symmetric matrix always is, with real eigenvalues (Strang, Introduction to Linear Algebra, chapter 6). When some eigenvalue is defective, the Jordan form takes the place of D: one block per eigenvector, its size read from the ranks of (A − λI)ᵏ.
Exact and numerical answers
The characteristic polynomial is computed in exact fractions by the Faddeev–LeVerrier recurrence (traces of A·Mₖ) and split into square-free factors, which gives the multiplicities exactly. Rational roots are then found and confirmed exactly, quadratic factors give roots of the form a ± b√D (complex when D < 0), and eigenvectors for these are computed by exact row reduction in those numbers.
A factor of degree 3 or more without rational roots or rational quadratic factors generally has no simple closed form, so its roots and their eigenvectors are given as decimals. Every eigenvalue is also computed independently by the QR algorithm — balancing, reduction to Hessenberg form and shifted QR sweeps with Francis double-shift steps, as described in Golub and Van Loan’s Matrix Computations — and the page shows how closely the two agree.
Limitations
- Square matrices up to 10 × 10 with rational entries (whole numbers, decimals, fractions). Irrational or complex entries are not accepted; type √2 as a decimal such as 1.41421356.
- Eigenvalues that are roots of an irreducible factor of degree 3 or more are shown as decimals (about 10 significant digits), and so are their eigenvectors; the cubic and quartic formulas are not used.
- When eigenvalues involve two different square roots (such as √2 and √3), the eigenvectors are exact but P and P⁻¹ are computed numerically.
- The Jordan form is given by its blocks; the change-of-basis matrix of generalised eigenvectors is not shown.
- Eigenvectors are only determined up to a non-zero factor: another calculator may show a multiple of the same vector.
Privacy
Everything happens in your browser. What you enter or open here is not uploaded or stored by MySmartCoPilot.
Frequently asked questions
What do I get without a pass?
Without a pass, Eigenvalue & Eigenvector Calculator shows the size of your matrix and what the full result holds, with the eigenvalues, the working, the polynomial, eigenvectors and PDP⁻¹ hidden. Until you unlock it, the result can’t be downloaded or copied. A Pro, Premium or Ultimate pass, a one-time payment that never renews, unlocks the full result. The pricing page lists the passes and their prices.
How do I find the eigenvalues of a 2 × 2 matrix?
Solve λ² − (a + d)λ + (ad − bc) = 0 for the matrix [[a, b], [c, d]]: the sum of the eigenvalues is the trace a + d and their product is the determinant ad − bc. For [[4, 1], [2, 3]] that is λ² − 7λ + 10 = 0, so λ = 5 and λ = 2.
How do I find an eigenvector once I have the eigenvalue?
Solve (A − λI)v = 0. Subtract λ from each diagonal entry, row-reduce, and write the solutions in terms of the non-pivot variables. For A = [[4, 1], [2, 3]] and λ = 5, A − 5I = [[−1, 1], [2, −2]] reduces to [[1, −1], [0, 0]], so v₁ = v₂ and v = (1, 1).
What do algebraic and geometric multiplicity mean?
Algebraic multiplicity counts how often λ is a root of the characteristic polynomial; geometric multiplicity counts how many independent eigenvectors it has. For [[1, 1], [0, 1]], λ = 1 has algebraic multiplicity 2 but only one eigenvector, so its geometric multiplicity is 1.
When is a matrix diagonalisable?
When it has n independent eigenvectors, that is when the geometric multiplicity of every eigenvalue equals its algebraic multiplicity. Then A = PDP⁻¹ with the eigenvectors as the columns of P. Matrices with n distinct eigenvalues and symmetric matrices are always diagonalisable.
Why do I get complex eigenvalues?
A real matrix can have a characteristic polynomial with no real roots, such as λ² + 1 for a 90° rotation. Its eigenvalues then come in conjugate pairs a ± bi, and the eigenvectors of a pair are conjugates of each other too.
Is det(A − λI) or det(λI − A) the characteristic polynomial?
Both are used: det(λI − A) always starts with +λⁿ, and det(A − λI) = (−1)ⁿ det(λI − A). They differ only in sign for odd n and have exactly the same roots. Choose the one your course uses in the options.