Eigenvalue and Eigenvector Calculator

Background.

This eigenvalue and eigenvector calculator takes a 2x2 or 3x3 matrix and returns everything the eigenvalue problem produces: the characteristic equation, every eigenvalue with its multiplicity, a basis of eigenvectors for each eigenspace, a straight yes-or-no verdict on whether the matrix is diagonalisable, and — when it is — the matrices P and D that satisfy A = PDP⁻¹. Type the matrix one row per line, with entries separated by commas or spaces, and read the whole picture at once.

An eigenvector of a square matrix A is a non-zero vector v that the matrix does not turn. Multiplying by A stretches or shrinks it, and may flip it end to end, but it stays on the same line through the origin: Av = λv. The scale factor λ is the eigenvalue attached to that direction. Almost every vector fails this test — a general matrix rotates almost everything it touches — so the handful of directions that survive tell you what the matrix fundamentally does. That is why eigenvalues show up wherever repeated application of the same linear step matters: the long-run behaviour of a Markov chain, the stability of an equilibrium in a differential equation, the principal axes of a stress tensor, the components in principal component analysis, and the natural frequencies of a vibrating structure.

The arithmetic starts from a rearrangement. Av = λv means (A − λI)v = 0, and a non-zero v can only solve that homogeneous system if the matrix (A − λI) is singular — if it collapses some direction to zero. A square matrix is singular exactly when its determinant vanishes, so the eigenvalues are precisely the roots of det(A − λI) = 0, the characteristic equation. For a 2x2 matrix that determinant expands to λ² − (trace)λ + (determinant), a quadratic. For a 3x3 it is a cubic. Once you have a root λ, you put it back into (A − λI)v = 0 and solve the homogeneous system by row reduction; the resulting null space is the eigenspace for that λ, and any basis for it is a valid set of eigenvectors.

Diagonalisation is the payoff, and it is where most of the confusion lives. If an n-by-n matrix has n linearly independent eigenvectors, you can stack them as the columns of a matrix P, put the matching eigenvalues on the diagonal of D, and write A = PDP⁻¹. That factorisation makes powers trivial — Aᵏ = PDᵏP⁻¹, and raising a diagonal matrix to a power just raises each diagonal entry — which is why it underpins so much of applied linear algebra. The common misconception is that a repeated eigenvalue rules diagonalisation out. It does not. This calculator reports two matrices with repeated eigenvalues that behave completely differently: one has a repeated eigenvalue with two independent eigenvectors and diagonalises fine, the other has a repeated eigenvalue with only one and cannot be diagonalised at all. The verdict on this page is computed from the eigenvector count, never from whether a root repeated.

Two limits are worth knowing before you read the result, and neither is buried in an FAQ. First, this page handles 2x2 and 3x3 matrices only. That is a mathematical boundary rather than a shortcut: the characteristic polynomial has the same degree as the matrix, and there is no formula in radicals for the roots of a general polynomial of degree five or higher, so every tool that claims to do large matrices is running an iterative numerical algorithm with its own convergence caveats. Rather than present an approximation as if it were exact, this calculator declines matrices larger than 3x3 and says so. Second, a real matrix can have complex eigenvalues — a rotation is the standard example, since no real line survives a rotation. When that happens the page reports the conjugate pair, reports that no real eigenvector exists, and says the matrix is diagonalisable over the complex numbers but not over the reals, instead of returning an empty answer.

What is eigenvalue and eigenvector calculator?

An eigenvector of a square matrix A is a non-zero vector v satisfying Av = λv for some scalar λ, and that λ is the corresponding eigenvalue. Rearranged, (A − λI)v = 0 has a non-trivial solution only when A − λI is singular, so the eigenvalues are exactly the roots of the characteristic equation det(A − λI) = 0. For a 2x2 matrix this expands to λ² − (tr A)λ + det A = 0; for a 3x3 matrix to λ³ − (tr A)λ² + c₁λ − det A = 0, where c₁ is the sum of the three principal 2x2 minors. The set of all solutions of (A − λI)v = 0 for a fixed λ is the λ-eigenspace, which is the null space Nul(A − λI); its dimension is the geometric multiplicity of λ, while the number of times λ repeats as a root of the characteristic polynomial is its algebraic multiplicity. The geometric multiplicity never exceeds the algebraic one. A matrix is diagonalisable — expressible as A = PDP⁻¹ with D diagonal — if and only if it has n linearly independent eigenvectors in total, equivalently if and only if the geometric and algebraic multiplicities agree for every eigenvalue. A matrix where they disagree is called defective. Two invariants give a free check on any answer: the sum of the eigenvalues always equals the trace of A, and their product always equals the determinant of A.

How to use this calculator.

1. 01Type your matrix into the box, one row per line. Use commas or spaces between entries — '4, 1' and '4 1' both work, and you can write a semicolon instead of a line break.
2. 02Enter 2 rows for a 2x2 matrix or 3 rows for a 3x3. Every row must have the same number of entries as there are rows, because eigenvalues only exist for square matrices.
3. 03Fractions are accepted directly: write 1/2 rather than 0.5 if that is how your problem is stated.
4. 04Read the eigenvalues at the top. They are listed with multiplicity, so a repeated root appears twice.
5. 05Check your answer for free: the eigenvalues should add up to the trace and multiply to the determinant, both of which are shown in the breakdown.
6. 06Read the eigenvectors panel for a basis of each eigenspace. Any non-zero multiple of a listed vector is an equally valid eigenvector.
7. 07Read the diagonalisable verdict. If it says yes, P and D are given directly and A = PDP⁻¹.
8. 08If the eigenvalues come back as a ± bi, the matrix rotates every real direction and has no real eigenvector — that is a genuine answer, not an error.

The formula.

Start from Av = λv and move everything to one side: (A − λI)v = 0. A homogeneous system has a non-zero solution only when its matrix is singular, and a square matrix is singular exactly when its determinant is zero, so every eigenvalue is a root of det(A − λI) = 0. Expanding that determinant for a 2x2 matrix gives λ² − (tr A)λ + det A, and for a 3x3 matrix gives λ³ − (tr A)λ² + c₁λ − det A, where c₁ adds up the three principal 2x2 minors. The 2x2 case is solved by the quadratic formula. The 3x3 case is solved in closed form by first removing the squared term with the substitution λ = t − p/3, which turns the cubic into t³ + Pt + Q; the sign of the discriminant Δ = (Q/2)² + (P/3)³ then decides the branch — three distinct real roots when Δ is negative (trigonometric form), a repeated root when Δ is zero, and one real root with a complex conjugate pair when Δ is positive (Cardano's form). Rounding happens in exactly two places, both after the mathematics is complete. Roots are computed at 60 significant digits and then polished by Newton's method on the exact characteristic polynomial; a root landing within 1e-9 of an integer is snapped to that integer before the eigenvector step, because a residual of even 1e-9 in λ makes (A − λI) non-singular and would wrongly return no eigenvector. Everything else is display formatting at 6 decimal places, or as an exact fraction where one with a denominator up to 1000 fits. With λ in hand, each eigenvector comes from row-reducing (A − λI) to reduced row echelon form and reading a null-space basis off the free columns. The diagonalisable verdict counts how many independent eigenvectors that produced in total and compares it to n — an unrounded integer comparison that no displayed value takes part in.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Wolfram MathWorld, "Eigenvalue" — defines the characteristic equation det(A − λI) = 0 and the characteristic polynomial. Retrieved 2026-07-29; open access.
2. [2]Margalit, D. & Rabinoff, J., Interactive Linear Algebra (Georgia Institute of Technology, 2019). §Eigenvectors gives the λ-eigenspace as Nul(A − λI); §Diagonalization states the theorem this page's verdict implements: an n × n matrix is diagonalisable if and only if it has n linearly independent eigenvectors. Retrieved 2026-07-29; open access.
3. [3]NIST Digital Library of Mathematical Functions, §1.11(iii) "Polynomials of Degrees Two, Three, and Four", equations 1.11.12–1.11.15 — the depressed-cubic substitution and root formulas used for the 3x3 characteristic polynomial. DLMF defines the discriminant as D = −4p³ − 27q², which is −108 times the Δ used in this module's code comments; the two never disagree about which branch applies. Retrieved 2026-07-29; open access.
4. [4]MIT OpenCourseWare, 18.06 Linear Algebra (Spring 2010), G. Strang — Unit III, Eigenvalues and Eigenvectors. Used as the independent check on the characteristic-equation derivation. Retrieved 2026-07-29; open access.
5. [5]Strang, G. (2016). Introduction to Linear Algebra, 5th ed., Wellesley–Cambridge Press. §6.1 "Introduction to Eigenvalues" and §6.2 "Diagonalizing a Matrix". Print reference — cited bibliographically, with no URL, rather than shipped as a link that would not resolve.