# QR decomposition

Put content here**Definition:** The **QR decomposition** of an $m \times n$ matrix $A$ (with $m \geq n$) is a factorization:
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$$A = QR$$
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where:
- $Q$ is an $m \times m$ orthogonal (unitary) matrix
- $R$ is an $m \times n$ upper triangular matrix (often written with zeros below the first $n$ rows)
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The **reduced QR** has $Q$ as $m \times n$ with orthonormal columns and $R$ as $n \times n$ upper triangular.
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**Example:**
$$\begin{pmatrix} 1 & 1 \\ 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{6} \\ 1/\sqrt{2} & -1/\sqrt{6} \\ 0 & 2/\sqrt{6} \end{pmatrix} \begin{pmatrix} \sqrt{2} & 1/\sqrt{2} \\ 0 & \sqrt{3/2} \end{pmatrix}$$
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**Methods:** Gram-Schmidt process, Householder reflections, or Givens rotations.
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**Applications:** solving least squares problems, eigenvalue algorithms (QR algorithm), and orthogonalization.

# Parents

* Factorization of matrices