Eigenvectors are directions a matrix only stretches, and eigenvalues are the stretch factors. They describe vibration modes, stability of systems, Google’s PageRank, quantum states and the principal components of data.
How to use the Eigenvalue and Eigenvector Calculator
- Choose the matrix size – the matching grid of fields appears.
- Enter the entries row by row (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The characteristic polynomial is solved numerically for the eigenvalues; the null space of A − λI gives the eigenvectors. Multiplicities are reported. The sum of the eigenvalues equals the trace and their product the determinant.
Formula
det(A − λI) = 0, (A − λI)v = 0
2 × 2: λ = tr/2 ± √(tr²/4 − det)
Worked example
Inputs — Matrix size: 2 × 2; a₁: 2; b₁: 1; a₂: 1; b₂: 2.
Results — Eigenvalues = λ = 1
λ = 3; Eigenvectors = λ = 1: (−1, 1)
λ = 3: (1, 1); Characteristic polynomial = det(A − λI) = λ² − 4λ + 3; Trace = 4; Determinant = 3.
Key terms
- Eigenvalue
- A scalar λ with Av = λv.
- Multiplicity
- How many times an eigenvalue is repeated.
Frequently asked questions
What is an eigenvalue?
A number λ such that Av = λv for some non-zero vector v.
Can eigenvalues be complex?
Yes – rotation matrices have complex eigenvalues.
How do eigenvalues relate to the trace and determinant?
Sum = trace, product = determinant.
Are eigenvectors unique?
Only up to scaling – any non-zero multiple is also an eigenvector.