The previous post discussed complex numbers, dual numbers, and double numbers. All three systems are constructed by adding some element to the real numbers that has some special algebraic property. The complex numbers are constructed by adding an element i such that i² = −1. The dual numbers add an element ε ≠0 with ε² = 0, and the double numbers are constructed by adding j ≠1 with j² = 1.
If adding special elements seems somehow illegitimate, there is an alternative way to define these number systems that may seem more concrete using 2 × 2 matrices. (A reader from 150 years ago would probably be more comfortable with appending special numbers than with matrices, but now we’re accustomed to matrices.)
The following mappings provide isomorphisms between complex, dual, and double numbers and their embeddings in the ring of 2 × 2 matrices.
Because the mappings are isomorphisms, you can translate a calculation in one of these number systems into a calculation involving real matrices, then translate the result back to the original number system. This is conceptually interesting, but it could also be useful if you’re using software that supports matrices but does not directly support alternative number systems.
You can also apply the correspondences from right to left. If you need to carry out calculations on matrices of the special forms above, you could move over to complex (or dual, or double) numbers, do your algebra, then convert the result back to matrices.
Functions of a matrix
The previous post looked at variations on Euler’s theorem in complex, dual, and double numbers. You could verify these three theorems by applying exp, sin, cos, sinh, and cosh to matrices. In each case you define the function in terms of its power series and stick in matrices. You should be a little concerned about convergence, but it all works out.
You should also be concerned about commutativity. Multiplication of real numbers is commutative, but multiplication of matrices is not, so you can’t just stick matrices into any equation derived for real numbers and expect it to hold. For example, it’s not true in general that exp(A + B) equals exp(A) exp(B). But it is true if the matrices A and B commute, and the special matrices that represent complex (or dual, or double) numbers do commute.