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It is well-known to many that $\mathbb{C}$ can be represented by matrices of the form $$\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$$ For example, see this question or this question. It is also discussed in the wikipedia article history of complex numbers article. Apparently, there is even an introductory complex variable textbook by Copson from 1935 which uses such matrices to define complex numbers. This is mentioned in Numbers by Ebbinghaus et. al. on page 69.

My question is simply this:

What is the history of this construction? Who first explained that complex numbers could be viewed as $2 \times 2$ matrices of the special form $\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$ ?

I realize this is just the regular representation of $\mathbb{C}$, and I realize such matrices are the matrices of a dilation composed with a rotation and possibly a reflection, but, the question still remains, who did found these first? References are appreciated.

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    $\begingroup$ I should point out that the instant you see that the action of multiplication by $a+bi$ is a linear map $\mathbb{R}^2 \to \mathbb{R}^2$ it's a fairly natural question to ask what the matrix is. $\endgroup$
    – jxnh
    Aug 4, 2014 at 5:58
  • $\begingroup$ Good Question, I first saw the approach in a series of lecture notes by Chritoph Thiele, but your reference predates mine. $\endgroup$
    – Mustafa Said
    Aug 4, 2014 at 9:35
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    $\begingroup$ I was a bit surprised to not find a reference in what I've read so far. Apparently, the idea that the complex numbers are identified with the plane traces back to Euler in 1749 and Cauchy in 1843 constructed them by calculating mod $x^2+1$ in $\mathbb{R}[x]$. These are discussed in "Numbers" which is a Springer text volume 123. $\endgroup$
    – James S. Cook
    Aug 4, 2014 at 13:28
  • $\begingroup$ Oops, I meant 1847 for Cauchy. $\endgroup$
    – James S. Cook
    Aug 4, 2014 at 13:36
  • $\begingroup$ From what I have read, the representation of complex numbers as points on the plane was introduced independently by Argand and Gauss around 1809. Can you please elaborate a bit on the Euler reference? $\endgroup$
    – timur
    Aug 12, 2014 at 1:49

2 Answers 2

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This set of lecture notes from Wedderburn explicitly says that a complex scalar $\alpha + i\beta$ can be written as \begin{equation} \left(\begin{array}{lr} \alpha & -\beta \\ \beta & \alpha \end{array}\right) \end{equation} on page 101 of the PDF (this is page 108 of the document when viewed in a PDF viewer). These notes are from 1934, which is obviously only slightly earlier than your example. However, the notes themselves are based on lectures given at Princeton starting in 1920, and it would seem that this notation goes back to 1907 because in that year Wedderburn (in his thesis) showed that associative hypercomplex systems can be represented by matrices. I've been unable to find his thesis online to check if this representation is explicitly written out, but I will update this post if I do.

Going back even further, in 1858 Arthur Cayley published "A Memoir on the Theory of Matrices" in which he mentions matrix representations of quaternions. Specifically, in item #45 on page 32 of the PDF (or on page 17 when viewed in a PDF viewer), he makes a passing mention of the fact that matrices $M$, $N$, and $L$ such that $L^2 = -1$, $M^2 = -1$, and $N = LM - -ML$ satisfy a system of equations that is the same as those that the quaternions satisfy. I didn't see anything in the above paper by Cayley about representing complex numbers with matrices, though I've seen a few passing references to Cayley coming up with the idea in 1858, so it may be the consensus of the mathematical community that the credit should go to Cayley.

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  • $\begingroup$ Well done. I've also started from Arthur Cayleys Memoir on the Theory of Matrices. Interesting papers from the 1870s about hypercomplex numbers and associative linear algebra from Caley and Pierce suggest, that the knowledge did already exist, but the notion seemed very unusual. In fact, I checked books from Kline, Dieudonne and found in Van der Waerdens History of Algebra a ref to the german encyclopedia of mathematics (written in 1898 - 1904). But the notation was not mentioned. I assume, earlier citations could be found in the English School, maybe some pupils from Caley. Regards. 1+ $\endgroup$
    – epi163sqrt
    Aug 15, 2014 at 10:26
  • $\begingroup$ Fantastic answer. These notes by Wedderburn are nicely formatted, they should help me with some other hypercomplex history questions. That said, the paper by Cayley made me laugh. I assign a half-dozen of his points as homework problems in linear algebra. That paper should be required reading for my linear algebra class. "Convertible matrices= commutative matrices". It's interesting to see the analysis so tied to linear systems as opposed to the algebra of matrices themselves. In any event, I have to agree the quaternion passage implicits complex number rep. matrices. Thanks! $\endgroup$
    – James S. Cook
    Aug 15, 2014 at 12:37
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Today was the first day of class in my complex analysis course. I sometimes attempt new derivations in real time to keep it fresh. Today, we got to the point of asking what was the reciprocal of $z=x+iy$. We said, let $w=a+ib$ and seek solutions of $wz=1$. This gives: $$ wz = (a+ib)(x+iy) = ax-by+i(bx+ay) = 1+i(0).$$ Equating real and imaginary parts reveals: $$ ax-by = 1 \qquad \& \qquad bx+ay = 0 $$ which is a system of linear equations which has matrix form: $$ \left[ \begin{array}{cc} x & -y \\ y & x \end{array}\right]\left[ \begin{array}{c} a \\ b \end{array}\right] =\left[ \begin{array}{c} 1 \\ 0 \end{array}\right] $$ We solve for $[a,b]^T$ by multiplying by the inverse of the $2 \times 2$ matrix for which we have the handy-dandy formula $\displaystyle \left[ \begin{array}{cc} x & -y \\ y & x \end{array}\right]^{-1} = \frac{1}{x^2+y^2}\left[ \begin{array}{cc} x & y \\ -y & x \end{array}\right]$. Thus, $$ \left[ \begin{array}{c} a \\ b \end{array}\right] = \frac{1}{x^2+y^2}\left[ \begin{array}{cc} x & y \\ -y & x \end{array}\right]\left[ \begin{array}{c} 1 \\ 0 \end{array}\right] = \frac{1}{x^2+y^2}\left[ \begin{array}{c} x \\ -y \end{array}\right].$$ Therefore, $a = \frac{x}{x^2+y^2}$ and $b = \frac{-y}{x^2+y^2}$ so $$\frac{1}{z}= \frac{x-iy}{x^2+y^2}.$$ I just found this a nice illustration of JHance's comment. In this routine calculation we stumble upon the $2 \times 2$ representations of both $z=x+iy$ and $1/z$. So, perhaps the real question to ask is not when the matrix representation was first given. Rather, the real question is simply when was the algebra of small matrices first known. I gather from yoknapatawpha's post of the 1858 paper of Cayley it may be a few years before that work. Apparently, the term matrix is Latin for "womb" and is due to Sylvester in 1850 as you may read at history of matrices. This makes me think there may exist some improvement on the Cayley answer.

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  • $\begingroup$ @JHance I thought this might amuse you. You're absolutely right. These $2 \times 2$ matrices just spring up in simple questions. $\endgroup$
    – James S. Cook
    Aug 19, 2014 at 3:10

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