##  [QR Decomposition](/qr-decomposition-0) 

 Definition

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

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

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

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

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

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

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

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

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.