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In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [ 1 ] [ 2 ] It is occasionally known as adjunct matrix , [ 3 ] [ 4 ] or "adjoint", [ 5 ] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose .
The definition via hom-sets makes symmetry the most apparent, and is the reason for using the word adjoint. The definition via counit–unit adjunction is convenient for proofs about functors that are known to be adjoint, because they provide formulas that can be directly manipulated. The equivalency of these definitions is quite useful.
Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type (Ax, y) = (x, By). Specifically, adjoint or adjunction may mean: Adjoint of a linear map, also called its transpose in case of matrices; Hermitian adjoint (adjoint of a linear operator) in functional analysis
In mathematics, specifically in operator theory, each linear operator on an inner product space defines a Hermitian adjoint (or adjoint) operator on that space according to the rule A x , y = x , A ∗ y , {\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,}
In category theory, a branch of mathematics, the formal criteria for adjoint functors are criteria for the existence of a left or right adjoint of a given functor. One criterion is the following, which first appeared in Peter J. Freyd 's 1964 book Abelian Categories, [ 1 ] an Introduction to the Theory of Functors :
Adjoint functors make the analogy between the tensor product and the smash product more precise. In the category of R -modules over a commutative ring R , the tensor functor ( − ⊗ R A ) {\displaystyle (-\otimes _{R}A)} is left adjoint to the internal Hom functor H o m ( A , − ) {\displaystyle \mathrm {Hom} (A,-)} , so that
The conjugate transpose "adjoint" matrix should not be confused with the adjugate, (), which is also sometimes called adjoint. The conjugate transpose of a matrix A {\displaystyle \mathbf {A} } with real entries reduces to the transpose of A {\displaystyle \mathbf {A} } , as the conjugate of a real number is the number itself.
That is, if is a function on the real line and is a self-adjoint operator, we wish to define the operator (). The spectral theorem shows that if T {\displaystyle T} is represented as the operator of multiplication by h {\displaystyle h} , then f ( T ) {\displaystyle f(T)} is the operator of multiplication by the composition f ∘ h ...