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adjoint endomorphism
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(Definition)
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Let $U$ be a finite-dimensional vector space over a field $\kfield$ and $B:U\times U\to\kfield$ 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 $T\adj:U\rightarrow U$ characterized by $$B(u,Tv) = B(T\adj u,v),\quad u,v\in U.$$
It is convenient to identify $B$ with a linear isomorphism $B:U\rightarrow U\dual$ in the sense that $$B(u,v) = (Bu)(v),\quad u,v\in U.$$ We then have $$T\adj = B^{-1} T\dual B.$$ To put it another way, $B$ gives an isomorphism between $U$ and the dual $U^*$ and the adjoint $T\adj$ is the endomorphism of $U$ that corresponds to the dual homomorphism $T\dual:U\dual\rightarrow U\dual$ Here is a commutative diagram to illustrate this idea: $$ \xymatrix{% U \ar[r]^{T^\star} \ar[d]^B & U \ar[d]^{B} \\ \;U^* \ar[r]^{T^*} & \;U^{*} } $$
Let $\bu_1,\ldots,\bu_n$ be a basis of $U$ and let $M\in \Mat_{n,n}(\kfield)$ be the matrix of $T$ relative to this basis, i.e. $$\sum_j M^j_{\,i}\, \bu_j = T(\bu_i).$$ Let $P\in\Mat_{n,n}(\kfield)$ denote the matrix of the inner product relative to the same basis, i.e. $$P_{ij} = B(\bu_i,\bu_j).$$ Then, the representing matrix of $T\adj$ relative to the same basis is given by $ P^{-1} M\supt P.$ Specializing further, suppose that the basis in question is orthonormal,
i.e. that $$B(\bu_i,\bu_j) = \delta_{ij}.$$ Then, the matrix of $T\adj$ is simply the transpose $M\supt$
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 $T\adj:U\rightarrow U$ by means of the familiar adjointness condition $$\langle u,Tv\rangle = \langle T\adj u,v\rangle,\quad u,v\in U.$$
However, the analogous operation at the matrix level is the conjugate transpose. Thus, if $M\in \Mat_{n,n}(\cnums)$ is the matrix of $T$ relative to an orthonormal basis, then $\overline{M\supt}$ is the matrix of $T\adj$ relative to the same basis.
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"adjoint endomorphism" is owned by rmilson.
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See Also: transpose
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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 8 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 9555 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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Pending Errata and Addenda
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