# matrix logarithm

Unlike the scalar logarithm, there are no naturally-defined bases for the matrix logarithm; therefore, the matrix logarithm is always taken to be the natural logarithm^{}. In general, there may be an infinite number of matrices $B$ satisfying $\mathrm{exp}(B)=A$; these are known as the logarithms of $A$.

As for the scalar natural logarithm, the matrix logarithm can be defined as a power series^{} when $A$ is a square matrix^{} and $$, where $||\cdot |{|}_{F}$ is the Frobenius matrix norm. The logarithm this formula produces is known as the *principal logarithm* of $A$.

$$\mathrm{log}(A)=-\sum _{k=1}^{\mathrm{\infty}}\frac{{(I-A)}^{k}}{k}=\mathrm{log}(I+X)=\sum _{k=1}^{\mathrm{\infty}}\frac{{(-1)}^{k+1}}{k}{X}^{k}$$ |

Since this series expansion does not converge for all $A$, it is not a global inverse function for the matrix exponential^{}. In particular, ${\mathrm{exp}}^{\mathrm{log}A}=A$ only holds for $$, and $\mathrm{log}({\mathrm{exp}}^{A})=A$ only holds for $$.

There are other, more general methods of calculating the matrix logarithm. For example, see \htmladdnormallinkAn Explicit Formula for the Matrix Logarithmhttp://arxiv.org/abs/math/0410556.

Title | matrix logarithm |
---|---|

Canonical name | MatrixLogarithm |

Date of creation | 2013-03-22 15:31:22 |

Last modified on | 2013-03-22 15:31:22 |

Owner | Andrea Ambrosio (7332) |

Last modified by | Andrea Ambrosio (7332) |

Numerical id | 11 |

Author | Andrea Ambrosio (7332) |

Entry type | Definition |

Classification | msc 15A90 |

Classification | msc 15A99 |

Related topic | NaturalLogarithm2 |

Related topic | MatrixFNorm |

Related topic | FrobeniusMatrixNorm |

Defines | principal logarithm |