Qrdecomposition Calculator
Our free fractions calculator solves qrdecomposition problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Qrdecomposition Calculator
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Formula: A = QR
Worked example — Q = [[0.7071, 0.4082], [0, 0.8165], [0.7071, -0.4082]], R = [[1.4142, 0.7071], [0, 1.2247]]
Formula
A = QR
Where Q is an orthogonal matrix (Q^T Q = I) with orthonormal columns computed via the Gram-Schmidt process, and R is an upper triangular matrix containing the projection coefficients and norms.
Worked Examples
Example 1: QR Decomposition of a 3x2 Matrix
Problem:Find the QR decomposition of A = [[1,1],[0,1],[1,0]].
Solution:Column 1: a1 = [1,0,1], ||a1|| = sqrt(2) q1 = [1/sqrt(2), 0, 1/sqrt(2)] r11 = sqrt(2) = 1.4142 Column 2: a2 = [1,1,0] r12 = q1 . a2 = 1/sqrt(2) = 0.7071 u2 = a2 - r12*q1 = [0.5, 1, -0.5] r22 = ||u2|| = sqrt(1.5) = 1.2247 q2 = u2/r22 = [0.4082, 0.8165, -0.4082]
Result:Q = [[0.7071, 0.4082], [0, 0.8165], [0.7071, -0.4082]], R = [[1.4142, 0.7071], [0, 1.2247]]
Example 2: QR of a Square 2x2 System
Problem:Find QR of [[3,1],[4,2]] to solve 3x+y=5, 4x+2y=6.
Solution:a1 = [3,4], ||a1|| = 5, q1 = [0.6, 0.8] r11 = 5, r12 = q1.[1,2] = 0.6+1.6 = 2.2 u2 = [1,2] - 2.2*[0.6,0.8] = [-0.32, 0.24] r22 = 0.4, q2 = [-0.8, 0.6] Solve Rx = Q^T b by back substitution
Result:R = [[5, 2.2], [0, 0.4]], Q = [[0.6, -0.8], [0.8, 0.6]]
Frequently Asked Questions
What is QR decomposition and what does it decompose a matrix into?
QR decomposition (also called QR factorization) breaks a matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R, written as A = QR. The matrix Q has orthonormal columns, meaning each column has unit length and all columns are mutually perpendicular. The matrix R is upper triangular, meaning all entries below the main diagonal are zero. For an m-by-n matrix with m greater than or equal to n, Q is m-by-n and R is n-by-n. This decomposition exists for every matrix and is one of the most important factorizations in numerical linear algebra, serving as the computational backbone for many algorithms including eigenvalue computation and least-squares solving.
How does the Gram-Schmidt process compute QR decomposition?
The Gram-Schmidt process is the classical algorithm for QR decomposition. It works by processing the columns of A one at a time, converting each into an orthonormal vector. For the first column, simply normalize it to get q1. For each subsequent column, subtract its projections onto all previously computed q vectors, then normalize the residual to get the next q vector. The R matrix entries are computed along the way: the diagonal entry r_kk is the norm of the residual before normalization, and the off-diagonal entry r_ik (for i less than k) is the dot product of q_i with the original column k. The modified Gram-Schmidt variant is numerically more stable because it subtracts projections one at a time rather than all at once.
What are the main applications of QR decomposition in numerical computing?
QR decomposition has numerous critical applications. The QR algorithm for computing eigenvalues repeatedly applies QR decomposition to iteratively converge to the eigenvalue structure of a matrix. It is the standard method used by LAPACK and MATLAB for eigenvalue computation. For solving least-squares problems Ax approximately equals b, QR decomposition provides a more numerically stable approach than the normal equations. In statistics, QR is used for fitting linear regression models. It is essential in computing the pseudoinverse and in rank-revealing factorizations. Modern applications include signal processing, data compression, and solving differential equations. The ubiquity of QR decomposition makes it one of the top ten algorithms of the twentieth century.
How does QR decomposition solve least-squares problems?
To solve the least-squares problem of minimizing the norm of Ax minus b, substitute A = QR to get QRx approximately equals b. Since Q has orthonormal columns, multiply both sides by Q-transpose to get Rx = Q-transpose times b. Because R is upper triangular, this system can be solved efficiently by back substitution. This approach is preferred over the normal equations (A-transpose A)x = A-transpose b because it avoids computing A-transpose A, which squares the condition number of A. The QR approach maintains the original condition number, providing much better numerical stability. For large-scale problems or ill-conditioned matrices, this improved stability can make the difference between getting meaningful results and getting numerical garbage.
What is the difference between thin QR and full QR decomposition?
For an m-by-n matrix with m greater than n, the full QR decomposition produces an m-by-m orthogonal Q and an m-by-n upper triangular R, where the bottom (m-n) rows of R are all zeros. The thin (or reduced) QR decomposition instead produces an m-by-n Q with orthonormal columns and an n-by-n upper triangular R. The thin version is computationally more efficient and uses less memory since it discards the columns of Q that correspond to the zero rows of R. Both decompositions contain the same essential information for solving linear systems and least-squares problems. In practice, the thin QR is used most often because the additional columns of Q in the full version are rarely needed.
How does Householder QR differ from Gram-Schmidt QR?
Householder QR uses orthogonal reflections (Householder transformations) instead of the projection-based Gram-Schmidt process. Each Householder transformation zeros out all elements below the diagonal in one column of the matrix, progressively building the upper triangular R while implicitly forming Q as the product of all the reflection matrices. This approach is numerically superior to classical Gram-Schmidt because it preserves orthogonality to machine precision, whereas classical Gram-Schmidt can lose orthogonality for ill-conditioned matrices. Householder QR requires approximately 2mn-squared minus (2/3)n-cubed floating point operations, which is about the same as Gram-Schmidt but with much better numerical properties. Most production linear algebra libraries use Householder QR as their default implementation.
What role does QR decomposition play in the QR algorithm for eigenvalues?
The QR algorithm is the most important method for computing eigenvalues of general matrices, and it uses QR decomposition as its core operation. Starting with matrix A0 = A, each iteration computes the QR decomposition A_k = Q_k R_k, then forms A_{k+1} = R_k Q_k (note the reversed order). Under certain conditions, the sequence A_k converges to an upper triangular matrix whose diagonal entries are the eigenvalues. In practice, shifts are applied (the shifted QR algorithm) to accelerate convergence dramatically, achieving cubic convergence rate. The matrix is also typically reduced to Hessenberg form first (upper triangular plus one subdiagonal) to reduce the cost of each QR step. This algorithm has been called one of the most important numerical algorithms ever developed.
How do you determine if a matrix has full column rank using QR decomposition?
A matrix has full column rank if and only if all diagonal entries of R in its QR decomposition are nonzero. Since R is upper triangular, its rank equals the number of nonzero diagonal entries. If any diagonal entry r_kk is zero, it means the k-th column of A is linearly dependent on the previous columns, and the rank is less than the number of columns. In practice, a diagonal entry is considered effectively zero if its absolute value is below a threshold related to machine precision and the matrix norms. Rank-revealing QR decompositions use column pivoting to ensure the diagonal entries of R decrease in magnitude, making rank determination more reliable for numerically rank-deficient matrices.
Can QR decomposition be applied to complex matrices?
Yes, QR decomposition extends naturally to complex matrices. For a complex matrix A, the decomposition gives A = QR where Q is a unitary matrix (Q times Q-conjugate-transpose equals the identity) rather than just orthogonal, and R is upper triangular with possibly complex entries. The Gram-Schmidt process works the same way but uses the complex inner product (conjugate-linear in the first argument). Householder reflections become Householder reflections using complex Householder vectors. The resulting Q matrix preserves the complex inner product structure. Complex QR decomposition is essential in signal processing, quantum mechanics, and any field where complex-valued data or operators arise naturally.
What is column pivoting in QR decomposition and why is it important?
Column pivoting adds a permutation step to QR decomposition, producing A times P = QR where P is a permutation matrix that reorders the columns. At each step of the factorization, the column with the largest remaining norm is selected as the next pivot column. This ensures that the diagonal entries of R are in decreasing order of absolute value, which has two major benefits. First, it provides a reliable way to estimate the numerical rank of the matrix by examining where the diagonal of R drops below a threshold. Second, it improves numerical stability for rank-deficient or nearly rank-deficient matrices. Column-pivoted QR is often used as a faster alternative to SVD when an approximate rank determination is sufficient for the application.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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