Complex Matrix Expansions

I was proud of this idea, as it was probably the best early research idea I’ve come up with.

2012-12-18

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 z = r\mathrm{e}^{\mathrm{i}\theta}, let \mathbf{Z} = r\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}. Some interesting properties follow immediately from this correspondence:

  1. If z = 0 then \mathbf{Z} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}. (zero matrix)
  2. If z = 1 then \mathbf{Z} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}. (identity matrix)
  3. z^{-1} corresponds to \mathbf{Z}^{-1}. (matrix inverse)
  4. z^\ast (complex conjugate) corresponds to \mathbf{Z}^\intercal. (matrix transpose)
  5. \det(\mathbf{Z}) = \lvert z\rvert^2. (matrix determinantabsolute value).

I liked to write (5) as  \lvert\mathbf{Z}\rvert = \lvert z\rvert^2, 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 a and b that correspond to matrices \mathbf{A} and \mathbf{B} respectively.

  • ab corresponds to \mathbf{A}\mathbf{B}. (homomorphic on multiplication)
  • a + b corresponds to \mathbf{A} + \mathbf{B}. (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 x + y\mathrm{i} would correspond to \begin{bmatrix}x & -y \\ y & x\end{bmatrix}.

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…

Suggestions

  1. Instead of expanding complex numbers into 2 x 2 real matrices, expand m x n complex matrices into 2m x 2n real matrices.
  2. Find analogues of every one of the above-mentioned properties under this generalized expansion. Warning: the analogue of (5) is not obvious!
  3. 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.

Bibliography

(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).

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s