QR Decomposition Calculator

QR decomposition factors a matrix A into the product A = QR, where Q is an orthogonal matrix (columns are orthonormal) and R is an upper triangular matrix. It is one of the most important matrix factorizations in numerical linear algebra.

The most common method is the Gram-Schmidt process, which orthogonalizes the columns of A one by one. Alternative methods include Householder reflections and Givens rotations, which are more numerically stable for large matrices.

Gram-Schmidt orthogonalizes a set of vectors by projecting each vector onto the space spanned by the previous ones and subtracting. For QR decomposition, it orthogonalizes the columns of A to form Q and records the projections in R.

QR decomposition is unique when A has linearly independent columns and the diagonal elements of R are restricted to be positive. Without this restriction, Q and —Q paired with R and —R both give valid decompositions.

Full QR: Q is m × m and R is m × n (with zero rows below). Reduced (thin) QR: Q is m × n and R is n × n. The reduced form is more compact and sufficient for most applications.

QR decomposition is used in solving least squares problems, computing eigenvalues (QR algorithm), finding matrix rank, solving linear systems, and is fundamental in many numerical algorithms.

For min‖Ax — b‖, compute A = QR, then Rx = Qᵀb. Since R is upper triangular, this is solved by back substitution. The solution is x = R⁻¹Qᵀb, which is the least squares solution.

The QR algorithm iteratively computes Aₖ = QₖRₖ and sets Aₖ₊₁ = RₖQₖ. The matrix converges to upper triangular form, and the diagonal elements converge to the eigenvalues. This is the standard method for computing all eigenvalues.

Householder reflections are orthogonal transformations that reflect a vector about a hyperplane. They are used as an alternative to Gram-Schmidt for QR decomposition, providing better numerical stability by avoiding accumulation of rounding errors.

For a positive definite matrix A, the Cholesky decomposition A = LLᵀ can be derived from QR: compute QR of A, then R = QᵀL where L is lower triangular. The Cholesky decomposition is essentially a special case of QR.

Gram-Schmidt requires O(mn²) operations. Householder reflections also require O(mn²) operations but are more stable. For an m × n matrix with m ≥ n, the complexity is O(mn²).

Since Q has orthonormal columns, QᵀQ = I. The Frobenius norm of Q is √n (where n is the number of columns), since each column has unit norm.

Yes, QR decomposition works for any m × n matrix. For m > n (overdetermined), it is particularly useful for least squares. For m < n (underdetermined), it can be used to find minimum norm solutions.

For an orthogonal matrix Q, det(Q) = ±1. If Q is a proper rotation (no reflections), det(Q) = 1. If Q includes an odd number of reflections, det(Q) = —1.

QR uses orthogonal and triangular factors (A = QR), while LU uses lower and triangular factors (A = LU). QR is more stable and works for non-square matrices, while LU is faster but requires pivoting for stability.

In regression analysis, QR decomposition is used to compute least squares estimates more stably than the normal equations. It is the method behind the lm() function in R and similar statistical software.

The condition number of an orthogonal matrix Q is always 1, since ‖Q‖₂ = 1 and ‖Q⁻¹‖₂ = ‖Qᵀ‖₂ = 1. This means Q does not amplify numerical errors.

For small matrices (2×2 or 3×3), hand computation is feasible using Gram-Schmidt. For larger matrices, the computation is best left to computers due to the large number of operations and potential for rounding errors.

Both QR and SVD decompose matrices into simpler factors. QR uses orthogonal and triangular factors, while SVD uses orthogonal and diagonal factors. SVD is more expensive but provides more information about the matrix.

Verify that: (1) A = QR, (2) QᵀQ = I (Q has orthonormal columns), (3) R is upper triangular. You can also check that ‖A‖ = ‖QR‖ and that the columns of Q are orthonormal.

Givens rotations are 2×2 orthogonal matrices that zero out individual elements. They are used as an alternative to Householder reflections for QR decomposition, especially on parallel computers where operations can be distributed.

The QR Decomposition Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The QR Decomposition Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.