I was proud of this idea, as it was probably the best early research idea I’ve come up with.
I became very interested in matrices and linear algebra after reading a paper on modelling origami using rotation and translation matrices (Belcastro and Hull, 2012). I began to play with expressing all kinds of things as matrices and seeing what “meanings” matrix operations had in those contexts.
Somehow complex numbers cropped up, and I decided that they were a good candidate for this “interpretation” because multiplying by a complex number meant a rotation and dilation of the complex plane – or an “amplitwist” (Needham, 1996). So I represented a complex number as a rotation matrix together with a scaling factor.
Given a complex number , let Some interesting properties follow immediately from this correspondence:
- If then (zero matrix)
- If then (identity matrix)
- corresponds to (matrix inverse)
- (complex conjugate) corresponds to (matrix transpose)
- (matrix determinant, absolute value).
I liked to write (5) as , using the “absolute value” notation for matrix determinants.
In terms of Abstract Algebra, the next two properties are more important. Suppose we have complex numbers and that correspond to matrices and respectively.
- corresponds to (homomorphic on multiplication)
- corresponds to (homomorphic on addition)
The multiplication property comes directly from how I defined the “matrix correspondence”: by treating matrix multiplication as a matrix transformation. The addition property was unexpected, but on hindsight it came from the fact that a complex number would correspond to
Prove all of the properties mentioned above. How would you generalize this “expansion” of a complex number further? My approach is in the Suggestions below…
- Instead of expanding complex numbers into 2 x 2 real matrices, expand m x n complex matrices into 2m x 2n real matrices.
- Find analogues of every one of the above-mentioned properties under this generalized expansion. Warning: the analogue of (5) is not obvious!
- Back then I had difficulty proving the analogue of (5) so I wrote a computer program to verify it by generating loads of randomized complex matrices and implementing arbitrary-precision arithmetic to multiply them.
(Belcastro and Hull, 2012) Sarah-marie Belcastro and Thomas C. Hull. Modelling the folding of paper into three dimensions using afﬁne transformations. Linear Algebra and its Applications 348, 273-282 (2002).
(Needham, 1996) Tristan Needham. Visual Complex Analysis. Oxford University Press (1996).