A matrix factorization A = Q R where Q is orthogonal (or unitary) and R is upper triangular; variants include full and reduced (thin) forms. It is used for solving linear systems, least-squares problems, and as a building block in numerical eigenvalue algorithms.

Principle

Separate a matrix into an orthonormal basis (Q) that changes coordinates and a triangular factor (R) that gives coordinates of original columns in that basis; orthogonality of Q preserves norm and stabilizes computations.

Demonstration

To solve min ||Ax - b||, compute A = QR, then R x = Q^T b (or R x = Q^* b in complex case) and solve the triangular system by back-substitution. Numerically, QR can be obtained via Householder reflectors, Givens rotations, or Gram–Schmidt procedures.

Misapplication

Using straightforward classical Gram–Schmidt without reorthogonalization on ill-conditioned matrices can produce highly nonorthogonal Q and inaccurate R. Assuming uniqueness of Q and R without fixing sign/phase or pivoting can be misleading, especially if A is rank-deficient.

Consequence

Provides numerically stable ways to solve least-squares and to orthonormalize columns; yields decompositions suitable for further numerical procedures like computing eigenvalues and performing rank-revealing factorizations.

Reversal

Use LU decomposition when a triangular times triangular factorization is acceptable (for square matrices and with pivoting), or use singular value decomposition (SVD) when explicit orthogonality and rank information are needed.

Boundary

Applies to matrices over real or complex inner-product spaces; Q is unique only up to orthonormal column signs/phases when R has no zero diagonals. QR does not directly apply to non-linear operators or infinite-dimensional operators without functional-analytic extension.

Semantic Tension

‘QR’ as a factorization versus ‘Gram–Schmidt’ as a process — QR denotes the matrix identity while Gram–Schmidt is a procedural route that may be unstable without modification.

Synthesis

QR decomposition expresses a matrix as an orthonormal change of basis followed by an upper-triangular coordinate map; implemented stably by Householder or Givens methods, it is central to solving linear and least-squares problems and to many numerical linear-algebra algorithms.