Mathematicians usually defined a linear function as a one to one correspondence between one value of the domain to one value of the range. A nonlinear function, on the other hand, is defined as many values of the domain corresponding to just one value of the range, which is really a many-to-one relation. However, a parameterized function is defined as one input to many outputs relation. These inputs and outputs can be functions or numbers. Nonetheless, the input to every input function is one single number called the parameter. This is most often the time parameter, t, of a given physical theory.

If a function f holds a simple relation to the positive time parameter, +tor f=t then its derivative with respect to t is unity: df/dt=1. If a function g is related to the negative time parameter, -t, then g=-t and its derivative is negative unity: dg/dt=-1. Furthermore, the product of these derivatives (df/dt)(dg/dt) = -1. This demonstrated that the directions of the time rate of change of both functions are always perpendicular in an abstract physical reality of the space-time background. This intrinsic perpendicularity is always implied by a square symmetric Hadamard matrix.