# adjoint endomorphism

## Definition (the bilinear case).

Let $U$ be a
finite-dimensional vector space^{} over a field $\mathbb{K}$, and $B:U\times U\to \mathbb{K}$ a symmetric^{}, non-degenerate bilinear mapping, for example
a real inner product^{}. For an endomorphism^{} $T:U\to U$ we
define the adjoint^{} of $T$ relative to $B$ to be the endomorphism
${T}^{\star}:U\to U$, characterized by

$$B(u,Tv)=B({T}^{\star}u,v),u,v\in U.$$ |

It is convenient to identify $B$ with a linear isomorphism $B:U\to {U}^{*}$ in the sense that

$$B(u,v)=(Bu)(v),u,v\in U.$$ |

We then have

$${T}^{\star}={B}^{-1}{T}^{*}B.$$ |

To put it another way, $B$ gives an
isomorphism^{} between $U$ and
the dual ${U}^{*}$, and the
adjoint ${T}^{\star}$ is the endomorphism of $U$ that corresponds to the
dual homomorphism (http://planetmath.org/DualHomomorphism)
${T}^{*}:{U}^{*}\to {U}^{*}$. Here is a commutative diagram^{} to
illustrate this idea:

$$\text{xymatrix}U\text{ar}{[r]}^{{T}^{\star}}\text{ar}{[d]}^{B}\mathrm{\&}U\text{ar}{[d]}^{B}{U}^{*}\text{ar}{[r]}^{{T}^{*}}\mathrm{\&}{U}^{*}$$ |

## Relation to the matrix transpose.

Let ${\mathbf{u}}_{1},\mathrm{\dots},{\mathbf{u}}_{n}$ be a basis of $U$, and let $M\in {Mat}_{n,n}(\mathbb{K})$ be the matrix of $T$ relative to this basis, i.e.

$$\sum _{j}{M}_{i}^{j}{\mathbf{u}}_{j}=T({\mathbf{u}}_{i}).$$ |

Let $P\in {Mat}_{n,n}(\mathbb{K})$ denote the matrix of the inner product relative to the same basis, i.e.

$${P}_{ij}=B({\mathbf{u}}_{i},{\mathbf{u}}_{j}).$$ |

Then, the representing matrix of ${T}^{\star}$ relative to the same basis is given by ${P}^{-1}{M}^{t}P.$ Specializing further, suppose that the basis in question is orthonormal, i.e. that

$$B({\mathbf{u}}_{i},{\mathbf{u}}_{j})={\delta}_{ij}.$$ |

Then, the matrix of ${T}^{\star}$ is
simply the transpose^{} ${M}^{t}$.

## The Hermitian (sesqui-linear) case.

If $T:U\to U$ is an endomorphism of a unitary space (a complex vector space equipped with a Hermitian inner product (http://planetmath.org/HermitianForm)). In this setting we can define we define the Hermitian adjoint ${T}^{\star}:U\to U$ by means of the familiar adjointness condition

$$\u27e8u,Tv\u27e9=\u27e8{T}^{\star}u,v\u27e9,u,v\in U.$$ |

However, the analogous operation^{} at the matrix level is the conjugate
transpose^{}. Thus, if $M\in {Mat}_{n,n}(\u2102)$ is the matrix of $T$
relative to an orthonormal basis^{}, then $\overline{{M}^{t}}$ is the
matrix of ${T}^{\star}$ relative to the same basis.

Title | adjoint endomorphism |
---|---|

Canonical name | AdjointEndomorphism |

Date of creation | 2013-03-22 12:29:36 |

Last modified on | 2013-03-22 12:29:36 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 12 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 15A04 |

Classification | msc 15A63 |

Synonym | adjoint |

Related topic | Transpose |

Defines | Hermitian adjoint |