QR Decomposition Calculator
Householder QR decomposition of a square matrix: orthogonal Q and upper-triangular R with A = Q·R.
Description
Factor a square matrix into an orthogonal matrix Q and upper-triangular matrix R.
QR decomposition expresses a matrix as the product of an orthogonal factor and an upper-triangular factor. It is widely used in least-squares methods, eigenvalue algorithms, and numerically stable linear algebra.
When to use QR Decomposition Calculator
- Inspect a Householder QR factorization
- Verify orthogonality and triangular structure
- Compare QR with LU decomposition for a square matrix
How the calculation works
Householder reflections successively zero entries below each diagonal position. The reflections accumulate into Q while transforming the input into R. Up to floating-point rounding, QᵀQ = I and A = Q·R.
A = Q · R and Qᵀ · Q = I Interpreting the result
Q's columns form an orthonormal basis and R is upper triangular. Signs are not unique: changing the sign of a column of Q and the matching row of R leaves their product unchanged, so valid software packages may return different-looking factors.
Important limitations
- This implementation accepts square matrices only, although QR is also defined for rectangular matrices.
- Very small residual entries may appear below R's diagonal because calculations use floating point.
- The returned signs follow the implementation's Householder convention.