Eigenvector Calculator

Work out eigenvector instantly with clear inputs, formula shown and shareable results.

About Eigenvector Calculator

Eigenvalues solve λ² − (trace)λ + det = 0. For each one, the eigenvector satisfies (A − λI)v = 0, which for a 2 × 2 matrix can be read off as (b, λ − a) whenever b is non-zero.

Formulas

Characteristic equation

λ² − (a + d)λ + (ad − bc) = 0

Eigenvector

v = (b, λ − a) when b ≠ 0

Worked example

With the default values below, Eigenvector Calculator returns Eigenvector for the larger eigenvalue of (1, 1). Change any input above and every figure here is recalculated in your browser.

Inputs used

Entry a: 4
Entry b: 1
Entry c: 2
Entry d: 3

Results

Eigenvector for the larger eigenvalue: (1, 1)
Larger eigenvalue: 5.00000000
Eigenvector for the smaller eigenvalue: (1, -2)
Smaller eigenvalue: 2.00000000

Step by step

1Values used
2Characteristic equation
3Eigenvector
4Eigenvector for the larger eigenvalue= (1, 1)
5Larger eigenvalue= 5.00000000
6Eigenvector for the smaller eigenvalue= (1, -2)
7Smaller eigenvalue= 2.00000000

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For [[4,1],[2,3]]?

The trace is 7 and the determinant 10, so the eigenvalues are 5 and 2, with eigenvectors (1, 1) and (1, −2) up to scale.

Are eigenvectors unique?

Only up to a scalar multiple. Any non-zero multiple of an eigenvector is also an eigenvector for the same eigenvalue.