QR Factorization is an essential matrix decomposition technique in linear algebra. It breaks a matrix A into the product of two matrices:
- Q, an orthonormal matrix (columns are orthogonal unit vectors)
- R, an upper triangular matrix
This decomposition helps simplify matrix calculations, making it easier to solve systems of equations, perform least squares regression, and handle numerical methods in engineering, physics, and data science.
Our free QR Factorization Calculator makes it easy for you to perform this decomposition online, no registration or installation required. Whether you’re a student verifying homework, a teacher preparing examples, or an engineer testing a matrix operation, this tool is built for you.
Simply:
- Choose the number of rows and columns for your matrix.
- Input the matrix values.
- Click “Calculate QR Decomposition” to instantly see the Q and R matrices, formatted and computed using the Gram-Schmidt orthogonalization process.
This tool saves time, reduces error, and makes linear algebra accessible to everyone.
🧮 Advanced QR Factorization Calculator
Select matrix size and enter values below to compute its QR decomposition.
QR Factorization isn’t just for textbooks, it’s used in real-world applications across science and tech:
- ✅ Solving overdetermined systems using least squares
- ✅ Improving stability in numerical algorithms
- ✅ Performing orthogonalization of vectors
- ✅ Calculating eigenvalues and eigenvectors
- ✅ Use in machine learning, signal processing, and scientific computing
The power of QR lies in its ability to break complex operations into manageable steps. For instance, in the least squares method, instead of solving the normal equations directly, QR decomposition simplifies the process into solving an upper triangular system, reducing computational errors.
Our QR Calculator Provides:
- ⚡ Instant and accurate results
- 🎯 Flexible matrix sizes (up to 4×4)
- 📱 Mobile-friendly interface
- 🧠 Zero ads and distractions
- 📘 Conceptual clarity with clean formatting
Whether you’re brushing up on linear algebra, building a model, or debugging code — this calculator is your smart assistant.
Still have questions? Here are answers to the most common ones:
What is QR decomposition used for?
It’s used in solving linear systems, least squares problems, and eigenvalue algorithms. It improves both performance and numerical accuracy.
How does this tool calculate Q and R?
We use the Gram-Schmidt process to orthogonalize the matrix columns, producing the Q matrix. Then we derive R by back-substituting using dot products.
Can I use decimals or negative numbers?
Yes, our calculator fully supports decimal and negative matrix entries.
Do I need to download anything?
No, everything runs in-browser. Just input your matrix and get results instantly.
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Use this QR Factorization Calculator anytime you’re working with matrices. Whether you’re solving equations, doing regression, or studying for exams — you’ll never have to solve matrix decompositions manually again.