Complex Numbers

Some people think of complex numbers as the points in the plane, in real 2-space. This is the approach Anton takes, which makes the definition of their multiplication a bit hard to motivate, and harder to remember.

Other people think of them as all formal expressions of the form a + bi with real a and b, subject to the familiar rules of arithmetic, along with the proviso that i² = –1. This approach seems palatable until you wonder if it isn’t all just chicken-scratches on paper.

Still others (yours truly among them) prefer a more concrete approach, as follows.

We already know all about matrix arithmetic for matrices of all shapes and sizes. Let’s reserve the term complex number for any 2x2 matrix

	[	a	b	]
A =
		–b	a

with both diagonal entries the same and each off-diagonal entry the negative of the other.

Quick checks (do them!) reveal that, whenever A and B are two complex numbers, so are:

(i) their ordinary matrix sum A + B;

(ii) their ordinary matrix difference A – B;

(iii) their ordinary matrix product AB;

(iv) the transpose A^T of the complex number A; and

(v) all the real multiples kA of the complex number A (here k may be any real number).

Moreover, despite the fact that, in general, the order of multiplication can make a big difference when multiplying 2x2 matrices (AC need not coincide with CA, as easy examples show – take, for example, any complex number A as above having b ≠ 0, and test with the matrix (not, in fact, a complex number)

	[	0	1	]
C =
		1	0

to discover just how different CA can, in general, be from AC), nonetheless

(vi) for complex numbers A and B one really will always have AB = BA (try it!).

Two more facts worth noting:

(vii) for A as above, det(A) = a² + b², and

(viii) for A as above, A^T A = det(A) Id_2x2 .

From (vii) it follows easily that

(ix) the only complex number A with det(A) = 0 is the zero matrix,

and from (viii) and (ix) it becomes easy to check that, unless A is the zero matrix, the only case in which det(A) has no real reciprocal,

(x) the matrix (1/det(A)) A^T serves as (two-sided) inverse for A.

How does this approach jibe with the first two approaches mentioned? Here’s how:

(1) The top row of a complex number “is” a point in the plane, and each point in the plane is the top row of one of our matrix-style complex numbers; and Anton’s seemingly ad hoc definition of multiplication for his complex numbers (cf. §10.1) is just what the top row of the matrix product of our matrix complex numbers becomes (likewise for sums, differences, real multiples – all behave similarly); and our transpose A^T simply has in its top row what Anton calls the complex conjugate of the top row of A.

(2) If we write 1 for Id_2x2 , and i for the matrix given by

	[	0	1	]
i =
		-1	0

then i² = –1 and, for the complex number given by the matrix A shown above, we have A = a1 + bi. So that alternate approach is not mere chicken-scratches on the paper.

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Prepared 19-20 Feb 2004.

Questions? Comments? Typos? Errata? Let me know, please.