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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 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.
The equation A x , y = x , A ∗ y {\displaystyle \langle Ax,y\rangle =\left\langle x,A^{*}y\right\rangle } is formally similar to the defining properties of pairs of adjoint functors in category theory , and this is where adjoint functors got their name from.
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
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In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [ 1 ] If A is a differentiable map from the real numbers to n × n matrices, then
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.
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