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Gram Schmidt Process Calculator

Work out gram schmidt process instantly with clear inputs, formula shown and shareable results.

About Gram Schmidt Process Calculator

Gram-Schmidt builds an orthogonal basis by subtracting from each vector its projection onto the ones already processed, then normalising. Dependent vectors reduce to zero and are dropped.

Formula

Gram-Schmidt step

wₖ = vₖ − Σⱼ<ₖ (vₖ·uⱼ / uⱼ·uⱼ)·uⱼ, then normalise

Worked example

With the default values below, Gram Schmidt Process Calculator returns Orthonormal basis of (0.948683, 0.316228) (-0.316228, 0.948683). Change any input above and every figure here is recalculated in your browser.

Inputs used

Vectors listed one after another
3, 1, 2, 2
Entries per vectoraccepts 2–6
2

Results

Orthonormal basis
(0.948683, 0.316228) (-0.316228, 0.948683)
Orthogonal basis before normalising
(3, 1) (-0.4, 1.2)
Vectors in the basis
2
Largest pairwise dot product (should be 0)
0.0000000000

Step by step

  1. 1Values used
  2. 2Gram-Schmidt step
  3. 3Orthonormal basis= (0.948683, 0.316228) (-0.316228, 0.948683)
  4. 4Orthogonal basis before normalising= (3, 1) (-0.4, 1.2)
  5. 5Vectors in the basis= 2
  6. 6Largest pairwise dot product (should be 0)= 0.0000000000

Assumptions and limits

  • The result is derived only from the 2 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 10 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 (3,1) and (2,2)?

The first normalises to (0.9486833, 0.3162278); the second loses its projection and becomes orthogonal to it.

Why does the largest dot product matter?

It is a numerical check: a genuinely orthonormal set has all pairwise dot products zero to within rounding error.

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