⊥ Gram-Schmidt Calculator
Gram-Schmidt Orthogonalization
Orthogonalize and orthonormalize any set of vectors. Enter column vectors as a matrix — each column is one vector. Full step-by-step projection subtraction shown.
Gram-Schmidt Orthogonalization
Each column of the input matrix is treated as a vector
✓ Step-by-Step✓ Normalized✓ Free⚡ Loading SymPy Engine…
ℹ Enter the input vectors as columns of the matrix. E.g., for 3 vectors in ℝ², enter a 2×3 matrix.
Input Vectors (columns)
Quick Examples
The Gram-Schmidt Process
The Gram-Schmidt process converts a set of linearly independent vectors into an orthonormal set (pairwise orthogonal unit vectors) that spans the same space.
Algorithm
- Start with v₁, v₂, ..., vₙ (the input vectors)
- Set u₁ = v₁
- For each vₖ: subtract projections onto all previous orthogonal vectors:
$$\mathbf{u}_k = \mathbf{v}_k - \sum_{j=1}^{k-1} \frac{\langle \mathbf{v}_k, \mathbf{u}_j \rangle}{\langle \mathbf{u}_j, \mathbf{u}_j \rangle} \mathbf{u}_j$$
- Normalize: eₖ = uₖ / ‖uₖ‖
Applications
- QR decomposition of matrices
- Finding orthonormal bases for subspaces
- Least squares problems
- Signal processing and data science (PCA)