QR decomposition
Undergraduate
Consider an $m \times n$ matrix $A$. A QR decomposition of $A$ is a factorization $A = Q R$, where the columns of $Q$ are orthonormal and $R$ is upper triangular.
Examples
Worked examples
- The matrix \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) : $Q = I$, $R = A$ Undergraduate
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The matrix \(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
: $Q = A$, $R = I$
Undergraduate
The matrix is already orthogonal.
Non-examples
Things that may be mistaken for examples.
-
A polar decomposition $A = Q S$, with $Q$ orthogonal and $S$ symmetric positive semidefinite
(definition)
Graduate
A polar decomposition $A = Q S$ is not a QR decomposition, because its factor $S$ is symmetric, not upper triangular.
Properties
- Every matrix has a QR decomposition. Applying the Gram–Schmidt process to the columns of $A$ produces one.
- QR decompositions are used to solve least-squares problems and to compute eigenvalues (the QR algorithm).
Distinctions involving this term
- Types of matrix decompositions: A matrix decomposition writes a matrix in terms of simpler matrices. Most are products: LU, PLU, and Cholesky decompositions use triangular factors, QR and singular value decompositions use orthogonal factors, and eigendecompositions, Schur decompositions, and Jordan normal forms are similarities $A = P M P^{-1}$ to a simpler matrix $M$ (diagonal, triangular, or made of Jordan blocks). In contrast, the Jordan–Chevalley decomposition is a sum of commuting diagonalizable and nilpotent parts, and the Toeplitz decomposition is a sum $H + K$ of a symmetric matrix $H$ and a skew-symmetric matrix $K$.