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Eigenvalue and Eigenvector Calculator (2×2 and 3×3, complex supported)

Characteristic polynomial, exact eigenvalues — rational, surd like (1±√5)/2, or complex like ±i for a rotation — with eigenvectors, algebraic and geometric multiplicities, and a clear verdict on diagonalizability.

Related tools

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Complex Number CalculatorAdd, subtract, multiply and divide complex numbers, with modulus, argument, conjugate and polar form.Angle Between Two Vectors Calculatorθ = arccos(a·b / (|a||b|)) in 2D or 3D, in degrees and radians, with the dot product, both magnitudes, the projections and a parallel / perpendicular / acute / obtuse verdict.Column Space Calculator (basis of Col(A), rank, pivot columns)A basis of the column space of any matrix up to 6×6, taken from the ORIGINAL pivot columns rather than the reduced ones — with the rank, the free columns, the full RREF and every row operation.Cramer's Rule CalculatorSolves a 2×2 or 3×3 system with exact fractions, shows every determinant expanded term by term, and when D = 0 separates the two cases most calculators merge: dependent with infinitely many solutions, or inconsistent with none.Cross Product Calculatora × b in 3D with the determinant expansion shown, the magnitude |a||b|sin θ, the parallelogram and triangle areas, the unit normal and the right-hand rule.Dot Product Calculatora·b in 2 to 10 dimensions with the term-by-term sum, both magnitudes as exact radicals, the angle in degrees and radians, the projections, and a figure. Parallel and orthogonal are decided in exact fractions, so (1,1,1)·(2,2,2) returns 0°, not NaN.Matrix CalculatorAdd, subtract and multiply matrices, scale, transpose, and find the determinant and inverse of A.LU Decomposition CalculatorFactor a square matrix into a lower L and upper U triangular matrix (Doolittle method).

Eigenvalue and Eigenvector Calculator (2×2 and 3×3, complex supported) is free to use as often as you like, directly from this page. You will find it under Matrices & vectors, with Complex Number Calculator and Angle Between Two Vectors Calculator for the neighbouring cases.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What is Eigenvalue and Eigenvector Calculator (2×2 and 3×3, complex supported)?

Characteristic polynomial, exact eigenvalues — rational, surd like (1±√5)/2, or complex like ±i for a rotation — with eigenvectors, algebraic and geometric multiplicities, and a clear verdict on diagonalizability.

When would I actually use this?

Checking a decomposition done by hand, solving a linear system, and finding the eigenvalues that describe how a transformation stretches space.

What is the most common mistake?

Inverting a matrix to solve a system. Elimination or a factorisation is both faster and far more stable numerically; an explicit inverse amplifies rounding error, especially when the matrix is close to singular.

How is Eigenvalue and Eigenvector Calculator (2×2 and 3×3, complex supported) different from Complex Number Calculator?

They sit next to each other but answer different questions: Complex Number Calculator is the one to open when you need it to add, subtract, multiply and divide complex numbers, with modulus, argument, conjugate and polar form. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Angle Between Two Vectors Calculator is the closest one after this: θ = arccos(a·b / (|a||b|)) in 2D or 3D, in degrees and radians, with the dot product, both magnitudes, the projections and a parallel / perpendicular / acute / obtuse verdict.

What else is worth having open alongside it?

Column Space Calculator (basis of Col(A), rank, pivot columns) and Cramer's Rule Calculator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The decompositions follow their textbook definitions with partial pivoting where it applies. Results are exact for small integer matrices and subject to floating-point rounding otherwise.

Further reading

All guides →
ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.ExplainerWhat Is a Z-Score? Standard Deviations Above the MeanA z-score is z = (x − μ) / σ — how many standard deviations a value sits from the mean. Learn how to compute it, read the standard normal curve, and turn it into a percentile.How-toHow to Find the Distance Between Two PointsUse the distance formula d = √((x₂−x₁)² + (y₂−y₁)²) to measure the straight-line distance between two points on a plane, with a worked example and the midpoint too.How-toHow to Calculate the Slope of a Line: Rise Over RunFind the slope of a line from two points with m = (y₂ − y₁) / (x₂ − x₁), and read what a positive, negative, zero, or undefined slope tells you.