Algebraic Conjugate

In order to more fully define the operation of division, we need to introduce a special operation called the algebraic conjugate.

The algebraic conjugate acts on elements of the algebra. An element that has been acted on by the algebraic conjugate operation is denoted by the symbol with a bar over the top, such as A. The algebraic conjugate is defined such that

AA = a

We don't really care yet what the value of a is, since we haven't really defined the conjugate, or the multiplication. What does matter is that a is a scalar. Using this definition of the algebraic conjugate, we can relate it to the "division" or inverse operation.

A^-1 = 1 / A = (1 / A) ( A / A ) = A / AA = (1 / a) A

Now, a is a real number, whose inverse is defined. This relationship between the inverse, the conjugate, and a is present in an identity from matrix algebra:

A^-1 = (1 / Det(A) ) Adjoint(A)

Thus, if we were using matrices to represent our physical quantities, a would be the determinant of the matrix and A would be the matrix Adjoint.

If we make the requirement that the conjugate operation should be linear, then we have the following property

A + B = A + B

We can use the relationship between the conjugate and the inverse to show that

AB = B A

Finally, since we know that taking an inverse twice should do nothing, we can again employ the relationship between the conjugate and the inverse to show that taking the conjugate twice also does nothing.