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Eigenvalues & Eigenvectors

Eigenvalues and eigenvectors calculator. Enter a square matrix A to compute the characteristic polynomial, all eigenvalues and their eigenvectors, with algebraic multiplicities.

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det(A−λI)=0, Av=λv
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4

📖 Tutorial｜Eigenvalues & Eigenvectors

1. Definition
If a number λ and nonzero vector v satisfy Av = λv, λ is an eigenvalue and v is an eigenvector.
2. Symbols
SymbolMeaning
λeigenvalue
veigenvector
A−λIcharacteristic matrix
3. How this tool works
  • Eigenvalues satisfy det(A−λI)=0;
  • For each λ solve (A−λI)v=0;
  • This tool solves 2×2 and 3×3 matrices numerically.
4. Steps
  1. Enter A;
  2. Click Calculate to see each λ and its v.
5. Example
Example: A=2 1;1 2. Characteristic equation (2−λ)²−1=0, λ₁=3, λ₂=1, eigenvectors (1,1) and (−1,1).
6. Pitfalls
Eigenvectors must be nonzero;
Complex eigenvalues come with complex eigenvectors;
Numerical results carry small error, shown to 4 digits.

❓ FAQ｜Eigenvalues & Eigenvectors

How to find eigenvalues?
Solve det(A−λI)=0.
How to find eigenvectors?
For each λ solve (A−λI)v=0.
Can an eigenvector be zero?
No, it must be nonzero.
What are complex eigenvalues?
When the characteristic equation has complex roots, eigenvectors are also complex.
Relation to trace and determinant?
tr(A)=sum of eigenvalues, det(A)=product of eigenvalues.