# What is the first usage of the term “Adjoint” and why was this word chosen?

The term "Adjoint" appears in many different mathematical areas and for sometimes seemingly different kinds of things. Wikipedia says -- "In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if $A$ is adjoint to $B$, then there is typically some formula of the type" $$\left( A x, y \right) = \left(x, B y\right)$$ Also, Wikipedia goes on to give examples and a few of them I repeat here:

• Hermitian Adjoint of a Linear Operator
• Adjoint Equation (as used in Differential Operators)
• Adjoint Matrix

And, there are other examples.

My question is about how and where was this term chosen. Better yet, who was first to start using it. Every time I see this word in a text or paper I am always thinking "There has to be a better more descriptive term to use".

## 2 Answers

Jeff Miller's very valuable collection of the origins of mathematical expressions has the entries adjoint equation (Lagrange), adjoint linear form (Cayley) and adjoint matrix (Bocher): http://jeff560.tripod.com/mathword.html

• I know of Jeff's collection web site but I totally forgot to make use of it. Thanks for the reminding pointer. In my own research I also found that this word adjoint whose dictionary definition seems to apply to Math topics only, exploded in usage in the 1950s. It is like the math community discovers more uses for this particular term and that spread in popularity like some kind new hip cultural expression. – K7PEH Apr 20 '17 at 14:29
• Does anyone know where exactly Lagrange used "adjoint equation"? Jeff Millers website doesn't seem to give a reference. – Michael Bächtold Jun 6 '18 at 12:25
• This page seems to indicate that it was 1766: sites.mathdoc.fr/cgi-bin/rbsm?cc=H__4_b_ – Jan Peter Schäfermeyer Jun 7 '18 at 16:50

In the article 267 of Disquisitiones Arithmeticae Gauss considers ternary form of second degree like $$ax^2+a'x'^2+a''x''^2+2bx'x''+2b'xx''+2b''xx'$$ and defines the adjoint to such form as $$Ax^2+A'x'^2+A''x''^2+2Bx'x''+2B'xx''+2B''xx'$$ where $$\begin{array}{lll} bb-a'a''=A,&b'b'-aa''=A',&b''b''-aa'=A'' \\ ab-b'b''=B,&a'b'-bb''=B',&a''b''-bb'=B'' \end{array}$$ Gauss clearly cannot speak about "adjoint to a matrix" (the term matrix in the usual sense was introduced after 1850), and moreover do not use 3x3 (symmetric) matrices to represent ternary forms (while uses something like 3x3 matrices to represent atransformation between forms) but it is clear that the matrix $$\left(\begin{array}{lll} A & B'' & B' \\ B'' & A' & B \\ B' & B & A'' \\ \end{array}\right)$$ is the opposite (this is due to the fact that Gauss defines the determinant of a form in such a way that it is actually the opposite of the determinant we use nowadays) of the adjoint to the matrix $$\left(\begin{array}{lll} a & b'' & b' \\ b'' & a' & b \\ b' & b & a'' \\ \end{array}\right)$$ which represents the original form. So it is possible that this is the first use in linear algebra of the term "adjoint" in a similar sense to the modern one.