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 $T:U\rightarrow U$ we define the adjoint of $T$ relative to $B$ to be the endomorphism $T^{\displaystyle\star}:U\rightarrow U$ , characterized by

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

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

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

We then have

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

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

\xymatrix{U\ar[r]^{T^{\star}}\ar[d]^{B}&U\ar[d]^{B}\\ \;U^{*}\ar[r]^{T^{*}}&\;U^{*}}

Relation to the matrix transpose.

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

\sum_{j}M^{j}_{\,i}\,\mathbf{u}_{j}=T(\mathbf{u}_{i}).

Let $P\in\mathop{\mathrm{Mat}}\nolimits_{n,n}(\mathbb{K})$ 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 $T^{\displaystyle\star}$ 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 $T^{\displaystyle\star}$ is simply the transpose $M^{t}$ .

The Hermitian (sesqui-linear) case.

If $T:U\rightarrow U$ 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 $T^{\displaystyle\star}:U\rightarrow U$ by means of the familiar adjointness condition

\langle u,Tv\rangle=\langle T^{\displaystyle\star}u,v\rangle,\quad u,v\in U.

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