QR Decomposition Calculator
MiniWebtool’s QR Decomposition Calculator uses Gram-Schmidt orthogonalization to factor supported 2×2 through 5×5 matrices into Q and R. Animated steps explain the projections, and an orthogonality check compares QᵀQ with the identity matrix.
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Enter a matrix below to get Q and R from the Gram-Schmidt steps, with a check that QᵀQ is the identity.
About QR Decomposition Calculator
The QR Decomposition Calculator factors any matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R, so that A = QR. Enter a 2×2 to 5×5 matrix (including non-square matrices where rows ≥ columns) and get the complete Gram-Schmidt orthogonalization with step-by-step solutions, interactive animation, orthogonality verification QᵀQ = I, and detailed educational insights.
At a Glance
QR Decomposition Calculator factors a matrix A into an orthogonal matrix Q and an upper triangular matrix R using Gram-Schmidt orthogonalization. It supports sizes from 2x2 to 5x5 and shows the steps.
| Feature | What you get | Good for |
|---|---|---|
| Matrix sizes | Enter a matrix from 2x2 to 5x5, including non-square matrices where rows are at least as many as columns. | Tall matrices from least-squares problems |
| Gram-Schmidt method | Each column is orthogonalized by subtracting projections onto earlier vectors, then normalized to build Q, with R upper triangular. | Learning how Q and R are produced |
| Animated steps | Play an animation and step through each column: original vector, projections removed, result, and normalized vector. | Following the process one column at a time |
| Orthogonality check | Verifies that QR equals A and that QᵀQ matches the identity matrix. | Confirming the factorization is correct |
| Preset examples | One-click presets: 2x2 Simple, 3x3 Full Rank, 3x2 Tall, Classic Example and 4x4 Matrix. | Trying the tool before typing your own matrix |
| Reference tables | Lists uses of QR such as least squares and eigenvalue algorithms, and compares QR with LU, Cholesky, SVD and eigendecomposition. | Choosing a decomposition for a task |
What Is QR Decomposition?
QR decomposition (also called QR factorization) writes a matrix A as:
$$A = QR$$
where Q is an orthogonal matrix (its columns are orthonormal vectors satisfying QᵀQ = I), and R is an upper triangular matrix. For an m×n matrix with m ≥ n and full column rank, the reduced QR gives Q as m×n and R as n×n.
The Gram-Schmidt Process Explained
Given column vectors a₁, a₂, …, aₙ of A, the classical Gram-Schmidt algorithm produces orthonormal vectors e₁, e₂, …, eₙ:
Step 1. Set u₁ = a₁, then normalize: e₁ = u₁ / ‖u₁‖.
Step 2. For each subsequent column aⱼ, subtract its projections onto all previous eₖ:
$$\mathbf{u}_j = \mathbf{a}_j - \sum_{k=1}^{j-1} (\mathbf{a}_j \cdot \mathbf{e}_k) \, \mathbf{e}_k$$
Then normalize: eⱼ = uⱼ / ‖uⱼ‖.
Step 3. The Q matrix has e₁, …, eₙ as columns. R is upper triangular with entries rᵢⱼ = eᵢ · aⱼ.
How to Use This Calculator
Step 1. Set the matrix dimensions (rows × columns). Rows must be ≥ columns for QR decomposition.
Step 2. Enter values into the grid, or click a quick example to load a preset. Use Tab or arrow keys to navigate.
Step 3. Click Decompose A = QR. The calculator runs the Gram-Schmidt process and displays Q and R.
Step 4. Watch the Gram-Schmidt animation to see how each column is orthogonalized: original vector → subtract projections → unnormalized result → normalized orthonormal vector.
Step 5. Verify the result: check that QR = A and QᵀQ = I (identity matrix). Step through the full derivation using the step navigator.
Applications of QR Decomposition
| Application | How QR Is Used |
|---|---|
| Least Squares (Ax ≈ b) | Solve Rx = Qᵀb by back-substitution — more stable than normal equations AᵀAx = Aᵀb |
| QR Algorithm for Eigenvalues | Repeatedly factor Aₖ = QₖRₖ, then set Aₖ₊₁ = RₖQₖ — converges to Schur form |
| Linear Systems (Ax = b) | Factor A = QR, then solve Rx = Qᵀb. More numerically stable than LU for ill-conditioned systems |
| Signal Processing | Adaptive beamforming and MIMO channel estimation use QR updates for real-time processing |
| Machine Learning | QR-based orthogonalization in neural network training, Gram-Schmidt in feature engineering |
QR vs. Other Matrix Decompositions
| Decomposition | Form | Best For |
|---|---|---|
| QR (this tool) | A = QR | Least squares, eigenvalue algorithms, numerically stable solves |
| LU | A = LU | Fast solves of square systems, determinant computation |
| Cholesky | A = LLᵀ | Symmetric positive definite systems (fastest) |
| SVD | A = UΣVᵀ | Rank analysis, pseudoinverse, PCA, image compression |
| Eigendecomposition | A = PDP⁻¹ | Matrix powers, differential equations, spectral analysis |
Frequently Asked Questions
What is QR decomposition?
QR decomposition factors a matrix A into the product of an orthogonal matrix Q (whose columns are orthonormal) and an upper triangular matrix R. Every real matrix with linearly independent columns has a unique QR factorization when we require R to have positive diagonal entries.
What is the Gram-Schmidt process?
The Gram-Schmidt process is an algorithm that takes a set of linearly independent vectors and produces an orthonormal set spanning the same subspace. It works by iteratively subtracting the projections onto all previously computed orthonormal vectors and then normalizing the residual.
Does QR decomposition work for non-square matrices?
Yes. For an m×n matrix where m ≥ n, the reduced (or thin) QR decomposition gives Q as m×n with orthonormal columns and R as n×n upper triangular. This is the form most commonly used in practice, especially for least squares problems.
When should I use QR instead of LU decomposition?
Use QR when numerical stability matters more than speed — for example, with ill-conditioned matrices, least squares problems, or eigenvalue computation. LU is faster (roughly 2× for square systems) but can amplify rounding errors. QR preserves vector norms because Q is orthogonal.
What is the difference between QR and SVD?
Both produce orthogonal factors, but SVD decomposes A into three matrices (UΣVᵀ) revealing singular values and rank, while QR gives two matrices (QR) and is faster to compute. SVD is preferred for rank-deficient problems and pseudoinverse computation; QR is preferred for solving full-rank systems and eigenvalue algorithms.
Reference this content, page, or tool as:
"QR Decomposition Calculator" at https://MiniWebtool.com/qr-decomposition-calculator/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-12
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