Understand the subject
What does Gram–Schmidt do?
Gram–Schmidt turns independent vectors into perpendicular directions by subtracting each earlier projection.
Use it to construct an orthogonal or orthonormal basis for the same span.
See the structure
What the calculation is doing
v₁v₂independent directions span a plane
Worked interpretation
Read the result in context
uₖ=vₖ−Σprojᵤⱼ(vₖ)
Interpret with care
Important boundary
Near-dependent numerical inputs can be sensitive; the page labels approximate normalization where needed.
Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.
Quick guide
How to use this calculator
- Set the matrix dimensions required by the task.
- Fill every visible entry and any operation-specific control.
- Read the result type, factors, classification, and disclosed limitations.
Calculation method
Apply the stated linear-algebra contract
uₖ = vₖ − Σⱼ<ₖ projᵤⱼ(vₖ)
Undefined, singular, dimensionally incompatible, dependent, or unsupported inputs receive an explicit message instead of a fabricated numeric result.
Worked example
Example interpretation
Normalization is approximate when a vector length is irrational.
uₖ = vₖ − Σⱼ<ₖ projᵤⱼ(vₖ)
Supported inputs
Precision and limits
Matrix size
Input matrices are limited to 1–6 rows and columns; operation-specific square, rank, or compatibility rules still apply.
Entry precision
Enter signed finite decimals with at most 30 digits and 15 decimal places.
Numerical policy
Results use real double-precision arithmetic, a 1e-10 structural tolerance where stated, bounded iterations, and labelled approximations.
