QR Decomposition Calculator
Decompose a matrix into Q (orthogonal) and R (upper triangular) matrices using Gram-Schmidt process.
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QR Decomposition
Every matrix A can be decomposed into the product of an orthogonal matrix Q and an upper triangular matrix R:
\[ A = QR \]
Properties:
- Q is orthogonal (Q^T Q = I)
- R is upper triangular
- Decomposition is unique if A is invertible
- Q has orthonormal columns
- R has positive diagonal elements
Calculation Methods:
- Gram-Schmidt Process
- Householder Transformations
- Givens Rotations
Applications:
- Solving linear systems
- Least squares problems
- Eigenvalue computation
- Matrix factorization
- Numerical stability
Geometric Interpretation:
- Q represents rotation/reflection
- R represents scaling
- Columns of Q form orthonormal basis
- R contains projection coefficients