QR Decomposition
Decompose a 2x2 matrix into orthogonal Q and upper triangular R via Gram-Schmidt. Free QR decomposition calculator for linear algebra, engineering and data science.
Enter the four elements of your 2×2 matrix A = [[a₁₁, a₁₂], [a₂₁, a₂₂]]
About This Calculator
The QR decomposition (also known as QR factorization) is a fundamental matrix factorization in linear algebra that writes a square or rectangular matrix A as the product of an orthogonal matrix Q and an upper triangular matrix R. For a 2×2 real matrix, this decomposition is computed using the Gram-Schmidt orthogonalization process applied to the columns of the matrix.
How QR Decomposition Works
For a 2×2 matrix A = [v₁ v₂] with column vectors v₁ and v₂, the Gram-Schmidt process proceeds in two stages. First, the first column v₁ is normalized to produce q₁ = v₁ / ||v₁||, and the entry r₁₁ is set to ||v₁||. Next, the second column v₂ is projected onto q₁ to compute r₁₂ = q₁ ⋅ v₂, and the orthogonal component w = v₂ − r₁₂ ⋅ q₁ is extracted. Finally, w is normalized to obtain q₂ = w / ||w||, with r₂₂ = ||w||. The result is Q = [q₁ q₂] and R = [[r₁₁, r₁₂], [0, r₂₂]], so that A = QR.
The calculator also handles the edge case where the first column is zero (producing an error message) and the case where the second column is linearly dependent on the first (singular matrix), where the corresponding column of Q is set to zero and the calculation remains numerically stable.
Applications of QR Decomposition
QR decomposition is one of the most important algorithms in numerical linear algebra. The QR algorithm is the standard method for computing eigenvalues and eigenvectors of general matrices. In data science and machine learning, QR decomposition powers linear regression by solving least-squares problems with superior numerical stability compared to normal equations. In signal processing, it is used in orthogonal matching pursuit and adaptive filtering. The decomposition is also widely used in control theory for solving Riccati equations, in computer graphics for camera pose estimation, and in GPS navigation algorithms.
Why Use This Calculator?
Manual computation of QR decomposition requires careful Gram-Schmidt orthogonalization with multiple dot products and normalizations, which is error-prone. This calculator automates the process, handles edge cases (zero columns, singular matrices), and displays both Q and R matrices to four decimal places. It is completely free, works offline after loading, and supports shareable URLs that preserve your input values for later reference.
Regional Notes
QR decomposition is a universal mathematical technique with no region-specific variations. The same formulas and algorithms apply worldwide in mathematics, engineering, physics, and computer science education and practice. The orthogonal and triangular factors are independent of currency, units, or regional conventions.
Frequently Asked Questions
What is QR decomposition?
QR decomposition (also called QR factorization) factors a matrix A into the product A = QR, where Q is an orthogonal matrix (QᵀQ = I) and R is an upper triangular matrix. For a 2×2 matrix, the Gram-Schmidt process orthogonalizes the columns of A to form Q, and the projection coefficients become the entries of R.
How does the QR decomposition work for a 2×2 matrix?
For a 2×2 matrix A = [v₁ v₂] with columns v₁ and v₂, the Gram-Schmidt process first normalizes v₁ to get q₁ = v₁/‖v₁‖ and sets r₁₁ = ‖v₁‖. It then projects v₂ onto q₁ to get r₁₂ = q₁ᵀv₂, subtracts to get the orthogonal component w = v₂ − r₁₂q₁, and normalizes w to get q₂ = w/‖w‖ with r₂₂ = ‖w‖. The result is Q = [q₁ q₂] and R = [[r₁₁, r₁₂], [0, r₂₂]].
What does it mean for a matrix to be orthogonal?
An orthogonal matrix Q is a square matrix whose columns are orthonormal (mutually perpendicular unit vectors). This means QᵀQ = QQᵀ = I (the identity matrix). Orthogonal matrices preserve lengths and angles under multiplication, making them essential in computer graphics, signal processing, and numerical linear algebra.
What are the applications of QR decomposition?
QR decomposition is fundamental in numerical linear algebra. It is used to solve linear least-squares problems (the backbone of linear regression in statistics and machine learning), compute eigenvalues via the QR algorithm, find matrix inverses, and solve systems of linear equations with better numerical stability than LU decomposition. It is also used in signal processing for orthogonal matching pursuit and in 3D graphics for camera calibration.
What is the difference between QR decomposition and LU decomposition?
QR decomposition factors A into an orthogonal matrix Q times an upper triangular R, while LU decomposition factors A into a lower triangular L times an upper triangular U. QR decomposition works for any matrix (including rectangular and singular matrices), whereas LU requires square matrices and can fail with zero pivots. QR is more numerically stable but computationally more expensive (O(n³) for both). For 2×2 matrices, both are equally efficient.
Can QR decomposition handle singular matrices?
Yes, QR decomposition works for singular matrices. If the second column of A is linearly dependent on the first, the Gram-Schmidt process will produce a zero vector after orthogonalization. In this case, the corresponding column of Q is set to zero and the R matrix has a zero on the diagonal. The decomposition A = QR still holds, and Q remains orthogonal. This is a key advantage over LU decomposition.
How is QR decomposition used in linear regression?
In linear regression, we solve the normal equations XᵀXβ = Xᵀy. Instead of directly inverting XᵀX, which can be numerically unstable, QR decomposition factorizes the design matrix X = QR. The least-squares solution reduces to Rβ = Qᵀy, which can be solved by back-substitution since R is upper triangular. This approach is preferred in statistical computing packages like R and MATLAB for its numerical stability.
Is this tool free?
Yes, all calculators on Calculy are completely free to use. There are no hidden fees, subscriptions, or usage limits. You can perform as many QR decompositions as you need.