Inputs
Example Data Table
A small example demonstrating a 3×2 basis and a 3×1 vector.
Example A
| 1 | 1 |
|---|---|
| 1 | -1 |
| 0 | 1 |
Example x
| 2 |
|---|
| 1 |
| 3 |
Use “Load Example” to paste these into the inputs.
Results
Fill the inputs and click Compute Projection to see results.
Formula Used
For a basis matrix A whose columns span a subspace of ℝm, the orthogonal projection onto Col(A) is:
P = A (AᵀA)−1 Aᵀ = Q Qᵀ, where A = Q R (QR).
We compute an orthonormal basis Q via modified Gram–Schmidt. This avoids explicit inversion and is typically more numerically stable. If requested, we attempt the normal-equations formula using (AᵀA)−1.
How to Use
- Enter your basis matrix A. Each row on a separate line.
- Optionally enter a vector x to project onto Col(A).
- Choose the method and number of decimal places to display.
- Click Compute Projection to generate P and optional P x.
- Use the CSV buttons to export P, Q, or P x as files.
- Click Download Results as PDF to capture the result card.
FAQs
A should be m×k, with columns representing basis vectors in ℝm. The projection matrix P is then m×m.
Dependent columns are automatically handled by QR: Q contains only independent directions. The rank r ≤ k determines the subspace dimension.
QR avoids inverting AᵀA, improving stability and handling near-singular bases better. It directly yields P = Q Qᵀ.
Yes. The complement projector is P⊥ = I − P. Projecting yields x⊥ = (I − P) x.
Ideally, ‖P² − P‖ and ‖Pᵀ − P‖ are near zero. Larger values suggest numerical issues or poorly scaled data.
Scaling affects numerical conditioning but not the subspace. QR mitigates issues; consider rescaling columns for tough cases.
Yes. After computing P once, apply it to each vector x. Or use Q to compute xproj = Q(Qᵀx) without forming P explicitly.