# self-dual

## Definition.

Let $U$ be a finite-dimensional inner-product space
over a field $\mathbb{K}$. Let $T:U\to U$ be an endomorphism,
and note that the adjoint endomorphism ${T}^{\star}$ is also an endomorphism
of $U$. It is therefore possible to add, subtract, and compare $T$
and ${T}^{\star}$, and we are able to make the following definitions. An
endomorphism $T$ is said to be self-dual (a.k.a. self-adjoint^{}) if

$$T={T}^{\star}.$$ |

By contrast, we say that the endomorphism is anti self-dual if

$$T=-{T}^{\star}.$$ |

Exactly the same definitions can be made for an endomorphism of
a complex vector space with a Hermitian inner product^{}.

## Relation to the matrix transpose.

All of these definitions have
their counterparts in the matrix setting. Let $M\in {Mat}_{n,n}(\mathbb{K})$ be the matrix of $T$ relative to an orthogonal
basis of $U$. Then $T$ is self-dual if and only if $M$ is a symmetric matrix^{},
and anti self-dual if and only if $M$ is a skew-symmetric matrix.

In the case of a Hermitian inner product we must replace the transpose^{}
with the conjugate transpose^{}. Thus $T$ is self dual if and only if $M$ is a Hermitian matrix, i.e.

$$M=\overline{{M}^{t}}.$$ |

It is anti self-dual if and only if

$$M=-\overline{{M}^{t}}.$$ |

Title | self-dual |

Canonical name | Selfdual |

Date of creation | 2013-03-22 12:29:40 |

Last modified on | 2013-03-22 12:29:40 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 5 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 15A63 |

Classification | msc 15A57 |

Classification | msc 15A04 |

Synonym | self-adjoint |

Related topic | HermitianMatrix |

Related topic | SymmetricMatrix |

Related topic | SkewSymmetricMatrix |

Defines | anti self-dual |