Gram-Schmidt Orthogonalization
The Gram-Schmidt Process
Algorithm
Given linearly independent vectors :
Step 1:
Step 2:
Step 3:
General Step:
Normalization
To get an orthonormal set, divide each orthogonal vector by its norm:
Properties
QR Decomposition
has columns (orthonormal vectors from Gram-Schmidt), and .
Example
Given :
Normalize:
Modified Gram-Schmidt
Applications
- QR algorithm for eigenvalue computation
- Least squares problems (QR factorization)
- Orthogonal polynomials (Legendre, Chebyshev, etc.)
- Orthonormal bases in function spaces
- Numerical linear algebra (more stable than normal equations)
Complexity
The classical Gram-Schmidt process requires operations for vectors in .