QR Factorization Calculator
Decompose matrices into orthogonal Q and upper triangular R matrices using Gram-Schmidt orthogonalization or Householder reflections. Enter your matrix to see step-by-step QR factorization with detailed orthogonalization process.
Input Matrix A
QR Factorization
Q Matrix (Orthogonal)
R Matrix (Upper Triangular)
Solve Least Squares Problem Ax = b
Understanding QR Factorization
Orthogonal Matrix Q
Q has orthonormal columns where Q^T Q = I. Each column represents an orthogonal basis vector for the column space.
Upper Triangular R
R contains the coefficients for expressing original columns as linear combinations of orthogonal basis vectors.
Gram-Schmidt Process
Classical orthogonalization method that sequentially orthogonalizes vectors using projection operations.
Householder Reflections
Numerically stable method using reflection matrices to introduce zeros below the diagonal systematically.
Mathematical Theory & Applications
What is QR Factorization?
QR factorization decomposes a matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R, where A = QR. This decomposition is fundamental in numerical linear algebra, providing a stable method for solving linear systems, least squares problems, and eigenvalue computations.
Basic Form: A = QR
Orthogonality: Q^T Q = I
Upper Triangular: R[i,j] = 0 for i > j
Uniqueness: Unique when R has positive diagonal
Historical Development
The QR decomposition builds upon the work of Jørgen Gram and Erhard Schmidt, who developed the Gram-Schmidt orthogonalization process in the early 1900s. The modern QR algorithm was refined by Alston Householder in the 1950s with his reflection method.
The development of numerically stable QR algorithms revolutionized computational linear algebra, making it possible to solve large-scale problems in engineering, physics, and data science with unprecedented accuracy and reliability.
Factorization Methods
Classical Gram-Schmidt
Sequential orthogonalization
Simple but numerically unstable
Good for educational purposes
Modified Gram-Schmidt
Improved numerical stability
Reorthogonalization at each step
Better for practical computation
Householder Reflections
Optimal numerical stability
Backward stable algorithm
Industry standard method
Givens Rotations
Sparse matrix friendly
Parallelizable operations
Specialized applications
Real-World Applications
Least Squares Regression
Solve overdetermined systems Ax=b by minimizing ||Ax-b||² using QR decomposition
Eigenvalue Computation
QR algorithm for finding eigenvalues through iterative QR factorizations
Signal Processing
Adaptive filtering, beamforming, and array signal processing applications
Computer Vision
Camera calibration, 3D reconstruction, and pose estimation problems
Machine Learning
Principal component analysis, linear regression, and neural network training
Numerical Analysis
Solving linear systems, matrix inversion, and condition number estimation
Control Systems
State estimation, Kalman filtering, and robust control design
Data Science
Dimensionality reduction, feature extraction, and statistical modeling
Frequently Asked Questions
Gram-Schmidt is conceptually simpler and orthogonalizes vectors sequentially, but can be numerically unstable. Householder reflections are more complex but provide superior numerical stability and are backward stable, making them preferred for practical computations.
Yes! QR factorization works for any m×n matrix with m≥n. For tall matrices (m>n), Q is m×n with orthonormal columns, and R is n×n upper triangular. This is particularly useful for least squares problems.
For overdetermined system Ax=b, QR factorization gives A=QR. The least squares solution is x = R⁻¹Q^T b. Since Q has orthonormal columns, Q^T Q = I, making the computation numerically stable and efficient.
QR factorization has O(mn²) time complexity for an m×n matrix. Householder method requires about 2mn²-2n³/3 operations, while Gram-Schmidt needs about mn² operations but with potential numerical issues.
Q has orthonormal columns, meaning Q^T Q = I. This preserves lengths and angles during transformations, making it numerically stable. Orthogonal matrices have determinant ±1 and their inverse equals their transpose.
The QR algorithm repeatedly applies QR factorization: A₀=A, then Aₖ₊₁=RₖQₖ where Aₖ=QₖRₖ. Under certain conditions, this sequence converges to a matrix with eigenvalues on the diagonal.
When columns are linearly dependent, the matrix is rank-deficient. QR factorization can still be computed, but R will have zero diagonal elements. Pivoting QR (column permutation) can reveal the rank structure more clearly.