BYTETOOLS

Eigenvalue and Eigenvector Calculator

Find eigenvalues and eigenvectors of 2×2 and 3×3 matrices from the characteristic polynomial, with algebraic and geometric multiplicity spelled out.

2×2
Size
7
trace(A) = Σλ
10
det(A) = Πλ
2
Real eigenvalues

Characteristic equation

λ² − (7)λ + (10) = 0

For a 2×2 matrix the characteristic polynomial is λ² − tr(A)λ + det(A), solved with the quadratic formula.

λ1 = 5

Algebraic multiplicity 1Geometric multiplicity 1

v1 = (1, 1)unit: (0.7071, 0.7071)

λ2 = 2

Algebraic multiplicity 1Geometric multiplicity 1

v1 = (1, -2)unit: (0.4472, -0.8944)

Summary

λ = 5 (algebraic 1, geometric 1)
  v1 = (1, 1)
λ = 2 (algebraic 1, geometric 1)
  v1 = (1, -2)

Check: the eigenvalues sum to the trace (7) and multiply to the determinant (10).

What is the Eigenvalue and Eigenvector Calculator?

The ByteTools Eigenvalue and Eigenvector Calculator takes a 2×2 or 3×3 matrix and finds the values λ for which A·v = λ·v has a non-zero solution.

  • Handles 2×2 and 3×3 matrices, including decimals and fractions like 3/4
  • Eigenvectors found as a null-space basis of A − λI, then scaled to whole numbers where possible
  • Reports algebraic against geometric multiplicity and flags defective matrices
  • Detects complex conjugate eigenvalue pairs instead of returning nothing
  • Trace and determinant cross-check on every result
  • Runs entirely in your browser — no uploads, works offline

How to use the Eigenvalue and Eigenvector Calculator

  1. 1

    Type your matrix in the text box: one row per line, values separated by spaces or commas.

  2. 2

    Read the characteristic equation the tool built from the trace, minors and determinant.

  3. 3

    Check each eigenvalue's card for its eigenvectors, plus the algebraic and geometric multiplicity.

  4. 4

    Look at the trace and determinant cross-check at the bottom to confirm the eigenvalues are consistent.

  5. 5

    Copy the full summary with the copy button if you need it in your notes.

About the Eigenvalue and Eigenvector Calculator

The ByteTools Eigenvalue and Eigenvector Calculator takes a 2×2 or 3×3 matrix and finds the values λ for which A·v = λ·v has a non-zero solution. A 2×2 matrix is solved through λ² − tr(A)λ + det(A) and the quadratic formula; a 3×3 matrix is solved through λ³ − tr(A)λ² + Mλ − det(A), where M is the sum of the three principal 2×2 minors, using the trigonometric form of Cardano's method so no complex cube roots are needed.

For every real eigenvalue the tool then row-reduces A − λI and reads a basis of its null space, which is exactly the eigenspace. It reports the algebraic multiplicity — how often λ repeats as a root — next to the geometric multiplicity, the number of independent eigenvectors it actually has, so defective matrices that cannot be diagonalised are called out rather than quietly mis-reported. Complex conjugate pairs are shown as a ± bi.

Everything runs as plain JavaScript inside your browser: the matrix is never uploaded and nothing is stored on a server. Every result carries a trace and determinant cross-check, because the eigenvalues must sum to the trace and multiply to the determinant.

Frequently asked questions

How do you find eigenvalues of a 2x2 matrix by hand?

Write the characteristic equation λ² − tr(A)λ + det(A) = 0, where tr(A) is the sum of the diagonal and det(A) is ad − bc, then solve that quadratic. For example [[4,1],[2,3]] gives λ² − 7λ + 10 = 0, so the eigenvalues are 5 and 2. This tool shows that same equation before it solves it.

What is the difference between algebraic and geometric multiplicity?

Algebraic multiplicity counts how many times an eigenvalue repeats as a root of the characteristic polynomial. Geometric multiplicity counts how many linearly independent eigenvectors it actually has. Geometric can be smaller — [[2,1],[0,2]] has λ = 2 twice but only one eigenvector direction.

Why does my matrix have complex eigenvalues?

A real matrix has complex eigenvalues when its characteristic polynomial has a negative discriminant, which happens whenever the transformation rotates rather than just stretches. A pure 90° rotation matrix [[0,−1],[1,0]] has eigenvalues ±i. Complex eigenvalues of a real matrix always come in conjugate pairs.

Can an eigenvalue be zero?

Yes, and it is meaningful: λ = 0 is an eigenvalue exactly when the matrix is singular, because det(A − 0·I) = det(A) = 0. The eigenvectors for λ = 0 are the null space of A. A matrix with a zero eigenvalue has no inverse.

Are eigenvectors unique?

No — any non-zero multiple of an eigenvector is still an eigenvector for the same eigenvalue, so only the direction is fixed. This calculator scales each eigenvector so its first non-zero entry is 1 and then clears small fractions, and also shows the unit-length version.

What does it mean if the matrix is defective?

A defective matrix has fewer independent eigenvectors than its size, so it cannot be written as PDP⁻¹ with a diagonal D. You need the Jordan normal form instead. The tool tells you this whenever a geometric multiplicity falls short of the algebraic one.

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