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
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The matrix \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)
: $Q = I$, $R = A$
Undergraduate
$A$ is upper triangular, so its LU decomposition is $I \cdot A$, and its QR decomposition is $I \cdot A$. $A$ is not diagonalizable, so its Jordan–Chevalley decomposition has a nonzero nilpotent part $N$.
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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.
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 $K$.