This demonstration application discusses several common matrix decompositions. A matrix decomposition is a factorization of a given matrix $A \in \mathbb{R}^{m \times n}$ into a product of matrices. Matrix decompositions are widely used to solve common problems in computer science as well as numerical analysis.

On this site we want to discuss some numerical methods for computing matrix decompositions implemented in PHP. The following table gives an overview of the decompositions covered here:

Decomposition | Factorizazion | Applicable for | Runtime |
---|---|---|---|

LU | $A = LU$ | $A \in \mathbb{R}^{n \times n}$, $A$ regular | $\mathcal{O}(\frac{1}{3}n^3)$ |

Cholesky | $A = LDL^T$ | $A \in \mathbb{R}^{n \times n}$, $A$ symmetric and positive definite | $\mathcal{O}(\frac{1}{6}n^3)$ |

QR: Givens Rotations | $A = QR$ | $A \in \mathbb{R}^{m \times n}$ | $\mathcal{O}(\frac{4}{3}n^3)$ |

QR: Householder Transformations | $A = QR$ | $A \in \mathbb{R}^{m \times n}$ | $\mathcal{O}(\frac{2}{3}n^3)$ |

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