I'm really confused about the terminology of adjoint.

Where in operator $A$ and $B$ was said to be adjoint iff $\langle Ax,y\rangle =\langle x,By\rangle$; yet in linear algebra adjoint of a matrix, which was also called adjugate of the matrix, was defined by cofactors.

Thus, in operator $adj(adj(A))=A$ because of the involutiveness.

Yet in linear algebra, (adjugate) $adj(adj(A^\prime))=det|A^\prime|^{n-2}A^\prime$ by computation.

Clearly, there seemed to be some connection between those two terminologies.

My questions were:

  1. What's the definitive connection?

  2. How am I suppose to avoid mistakes, and distinguish the usage of those two terminologies?

*Can assume $A$ is a bounded linear operator from a Hilbert space, definition as given.

Notes: I saw a previous example Adjoint and Adjugate are same or different? where it was answered in a way such that the definition was different. However, since operator could be presented in its matrix form and matrix could be seen as an operator, I'd like to know:

  1. Under which minium circumstance those two terminologies were equivalent.

  2. If there is any way to present one of the terminologies in terms of the combination of another. For example, a silly case in matrix form when $A$ was constant $Adjugate(Adjugate(A))=det |A|^{n-2}A=det |A|^{n-2}adj(adj(A))$


The choice of the word adjoint for adjugate in linear algebra is fairly unfortunate. My book refers to the adjugate as the “classical adjoint” to distinguish it from the more modern meaning of adjoint. Thus, if you want to consider an adjoint from an operator perspective, the corresponding object in linear algebra is the transpose not the adjugate. It satisfies involutiveness and you can check that it satisfies the criteria you stated. The only place I can see the classical and modern adjoint in linear algebra coinciding are in the orthogonal matrices!


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