Gram-Schmidt Calculator
Orthogonalize and orthonormalize vectors using the Gram-Schmidt process. Get step-by-step solutions, visualizations, and perform QR decomposition with detailed mathematical explanations.
Vector Input
Gram-Schmidt Process Theory
What is the Gram-Schmidt Process?
The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space ℝⁿ. It takes a finite, linearly independent set of vectors and generates an orthogonal set that spans the same subspace.
Orthogonalization Formula:
u₁ = v₁
u₂ = v₂ - proj_{u₁}(v₂)
u₃ = v₃ - proj_{u₁}(v₃) - proj_{u₂}(v₃)
...
Projection Formula: proj_u(v) = (⟨v,u⟩/⟨u,u⟩)u
Algorithm Steps
Step 1: Initialize
Set the first orthogonal vector u₁ = v₁
Step 2: Project and Subtract
For each subsequent vector vₖ, subtract all projections onto previous orthogonal vectors
Step 3: Normalize (Optional)
Divide each orthogonal vector by its magnitude to get orthonormal vectors
Step 4: Verify
Check that all resulting vectors are orthogonal (dot product = 0)
Applications & Importance
QR Decomposition
Foundation for QR matrix factorization used in solving linear systems
Least Squares
Essential for solving overdetermined systems and regression analysis
Signal Processing
Creating orthogonal basis functions for signal decomposition
Computer Graphics
Generating orthonormal coordinate systems for 3D transformations
Machine Learning
Principal Component Analysis and feature orthogonalization
Numerical Analysis
Improving numerical stability in various algorithms
Worked Examples
Example 1: 2D Vectors
Input Vectors:
v₂ = [1, 2]
Solution:
Step 1: u₁ = v₁ = [3, 4]
Step 2: u₂ = v₂ - proj_{u₁}(v₂) = [1, 2] - (11/25)[3, 4] = [-8/25, 18/25]
Orthogonal basis: {[3, 4], [-8/25, 18/25]}
Orthonormal basis: {[3/5, 4/5], [-4/5, 3/5]}
Example 2: 3D Vectors
Input Vectors:
v₂ = [1, 0, 1]
v₃ = [0, 1, 1]
Solution:
Step 1: u₁ = [1, 1, 0]
Step 2: u₂ = [1, 0, 1] - (1/2)[1, 1, 0] = [1/2, -1/2, 1]
Step 3: u₃ = [0, 1, 1] - (1/2)[1, 1, 0] - (1/3)[1/2, -1/2, 1] = [-1/3, 2/3, 2/3]
Result: Three orthogonal vectors spanning ℝ³
Frequently Asked Questions
Orthogonal vectors are perpendicular to each other (dot product = 0), while orthonormal vectors are orthogonal AND have unit length (magnitude = 1). The Gram-Schmidt process can produce either, depending on whether you normalize the results.
If the input vectors are linearly dependent, the Gram-Schmidt process will produce a zero vector at some step. This indicates that the corresponding input vector can be expressed as a linear combination of the previous vectors.
The Gram-Schmidt process is the foundation of QR decomposition. The Q matrix contains the orthonormal vectors produced by Gram-Schmidt, while the R matrix contains the coefficients used in the orthogonalization process.
Yes! The Gram-Schmidt process works in any finite-dimensional inner product space. This calculator supports up to 5D vectors, but the mathematical principle extends to any dimension.
The classical Gram-Schmidt process can be numerically unstable for nearly linearly dependent vectors. In practice, the modified Gram-Schmidt process or Householder reflections are often preferred for better numerical stability.
Check that: (1) All pairs of orthogonal vectors have dot product ≈ 0, (2) Orthonormal vectors have magnitude ≈ 1, (3) The span of the output vectors equals the span of the input vectors.