PlanetMath: adjoint endomorphism

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adjoint endomorphism

(Definition)

Definition (the bilinear case).

Let

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

relative to

to be the endomorphism $T^{\displaystyle \star}:U\rightarrow U$ , characterized by

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

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

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

We then have

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

To put it another way,

gives an isomorphism between

and the dual

, and the adjoint $T^{\displaystyle \star}$ is the endomorphism of

that corresponds to the dual homomorphism $T^*:U^*\rightarrow U^*$ . Here is a commutative diagram to illustrate this idea:

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{% U \ar[r]^{T^\star} \ar[d]^B & U \ar[d]^{B} \ \;U^* \ar[r]^{T^*} & \;U^{*} } } \end{xy}$

Relation to the matrix transpose.

Let $\mathbf{u}_1,\ldots,\mathbf{u}_n$ be a basis of

, and let $M\in \mathop{\mathrm{Mat}}\nolimits _{n,n}(\mathbb{K})$ be the matrix of

relative to this basis, i.e.

$\displaystyle \sum_j M^j_{\,i}\, \mathbf{u}_j = T(\mathbf{u}_i).$

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.

$\displaystyle P_{ij} = B(\mathbf{u}_i,\mathbf{u}_j).$

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

$\displaystyle B(\mathbf{u}_i,\mathbf{u}_j) = \delta_{ij}.$

Then, the matrix of $T^{\displaystyle \star}$ is simply the transpose

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). In this setting we can define we define the Hermitian adjoint $T^{\displaystyle \star}:U\rightarrow U$ by means of the familiar adjointness condition

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

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 relative to an orthonormal basis, then $\overline{M^t}$ is the matrix of $T^{\displaystyle \star}$ relative to the same basis.

"adjoint endomorphism" is owned by rmilson.

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See Also: transpose

Other names:

adjoint

Also defines:

Hermitian adjoint

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Cross-references: orthonormal basis, conjugate transpose, level, operation, adjointness, complex, unitary space, transpose, orthonormal, matrix, basis, commutative diagram, isomorphism, linear isomorphism, endomorphism, inner product, real, bilinear mapping, non-degenerate, symmetric, field, vector space, finite-dimensional
There are 17 references to this entry.

This is version 9 of adjoint endomorphism, born on 2002-02-26, modified 2006-03-19.
Object id is 2718, canonical name is AdjointEndomorphism.
Accessed 8203 times total.

Classification:

AMS MSC:	15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
	15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)

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