adjoint endomorphism

Definition (the bilinear case).

Let $U$ be afinite-dimensional vector space over a field $\\mathbb{K}$ , and $B:U\\times U\\to\\mathbb{K}$ a symmetric, non-degenerate bilinear mapping, for examplea real inner product. For an endomorphism $T:U\\rightarrow U$ wedefine 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 anisomorphism between $U$ andthe dual $U^{*}$ , and theadjoint $T^{\\displaystyle\\star}$ is the endomorphism of $U$ that corresponds to thedual homomorphism (http://planetmath.org/DualHomomorphism) $T^{*}:U^{*}\\rightarrow U^{*}$ . Here is a commutative diagram toillustrate 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 $M\\in\\mathop{\\mathrm{Mat}}\olimits_{n,n}(\\mathbb{K})$ be the matrix of $T$ relative to this basis, i.e.

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

Let $P\\in\\mathop{\\mathrm{Mat}}\olimits_{n,n}(\\mathbb{K})$ denote the matrix of the inner productrelative 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 basisis given by $P^{-1}M^{t}P.$ Specializing further, suppose that thebasis in question is orthonormal, i.e. that

B(\\mathbf{u}_{i},\\mathbf{u}_{j})=\\delta_{ij}.

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

The Hermitian (sesqui-linear) case.

If $T:U\\rightarrow U$ is an endomorphism of a unitary space (a complexvector space equipped with a Hermitian inner product (http://planetmath.org/HermitianForm)). In this setting we can define we definethe Hermitian adjoint $T^{\\displaystyle\\star}:U\\rightarrow U$ by means of the familiaradjointness 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 conjugatetranspose. Thus, if $M\\in\\mathop{\\mathrm{Mat}}\olimits_{n,n}(\\mathbb{C})$ is the matrix of $T$ relative to an orthonormal basis, then $\\overline{M^{t}}$ is thematrix of $T^{\\displaystyle\\star}$ relative to the same basis.