When we introduced the concept of graphing complex numbers on the complex plane, we did so from the point of view of considering complex numbers as 2-dimensional vector spaces over R. The main point of this was to introduce polar notation for complex numbers, but we first noted that addition of complex numbers followed the usual parallelogram rule for vector addition, and I made mention of the fact that there was a geometrical interpretation of complex multiplication as well.
It is at this point that we want to go back and figure out just what that geometrical interpretation of multiplication is. The inspiration comes from the polar definition of complex number multiplication: (r1(e-to-the-(i(theta1))))(r2(e-to-the-(i(theta2)))) will equal (r1r2)(e-to-the-(i(theta1 + theta2))).
Let us consider for a moment the case when r1 and r2 are both 1. Well, then we are looking at (e-to-the-(i(theta1)))(e-to-the-(i(theta2))) = e-to-the-(i(theta1 + theta2)). In this case, to multiply our complex numbers, we simply need to add their arguments. Another way to think of this is that we would start at e-to-the-(i(theta1)) and rotate an additional theta2 around the origin to get to e-to-the-(i(theta1 + theta2)). So that is to say that multiplying by e-to-the-(i(theta2)) is the same as rotating by theta2.
If we let r1 and r2 be numbers other than 1, the situation becomes only a bit more complicated. First of all, we still see that we get r1(e-to-the-(i(theta1 + theta2))) from r1(e-to-the-(i(theta1))) by rotating by theta2, and then to get from r1(e-to-the-(i(theta1 + theta2))) to (r1r2)(e-to-the-(i(theta1 + theta2))), we simply need to multiply by the scalar r2, which is the same thing as dilating (or contracting, if r2 is less than 1) by r2.
So we see that, geometrically, we can think of multiplication by a complex number as a combination of a rotation and a contraction. As these actions are both linear mappings over R2, their composition is also a linear mapping, and as such, there must be a matrix in M(2, 2) that represents it.
To find this matrix, let’s fix a complex number a = a + ib, or equal to the vector [a; b], and we’ll define the linear mapping La from R2 to R2 by La([x; y]) will equal [ax – by; bx + ay]. Now, since (a + bi)(x + yi) = (ax – by) + (bx + ay)i, this is the linear mapping for multiplying [x; y] by [a; b]. To find its standard matrix, we first need to see that L_alpha([1; 0]) = [a; b], and L_alpha([0; 1]) = [-b; a]. From this, we get that bracket-[L_alpha] is the matrix [a, -b; b, a].
Let’s look at an example. Let’s let alpha be the complex number 3 – 4i, and let’s find [M_alpha] and use it to calculate (3 – 4i)(2 + 5i). Well, we have that [M_alpha] will equal [a, -b; b, a], which, in this case, will make it [3, +4; -4, 3]. And thus, the product (3 – 4i)(2 + 5i) can be thought of as the matrix [M_alpha] times the vector [2; 5], and if we perform the calculations, we get the vector [26; 7], which we equate with the complex number 26 + 7i.