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Complex Numbers, Vectors and Matrices

Module by: Steven Cox

Summary: This module sets out to instruct about complex numbers: what they are, what they mean, how to manipulate them, and the different ways to describe them (i.e. polar form). The second half of this module proposes to introduce the characteristics of complex vectors and matrices and how they compare to the laws governing standard vectors and matrices.

Complex Numbers

A complex number is simply a pair of real numbers. In order to stress however that the two arithmetics differ we separate the two real pieces by the symbol +ⅈ . More precisely, each complex number, zz, may be uniquely expressed by the combination x+ⅈy x y , where xx and yy are real and ⅈ denotes -1 -1 . We call xx the real part and yy the imaginary part of zz. We now summarize the main rules of complex arithmetic.

If z1=x1+ⅈy1 z1 x1 y1 and z2=x2+ⅈy2 z2 x2 y2 then

Definition 1: Complex Addition
z1+z2≡x1+x2+ⅈy1+y2 z1 z2 x1 x2 y1 y2
Definition 2: Complex Multiplication
z1z2≡x1+ⅈy1x2+ⅈy2=x1x2-y1y2+ⅈx1y2+x2y1 z1 z2 x1 y1 x2 y2 x1 x2 y1 y2 x1 y2 x2 y1
Definition 3: Complex Conjugation
z1¯≡x1-ⅈy1 z1 x1 y1
Definition 4: Complex Division
z1z2≡z1z2z2¯z2¯=x1x2+y1y2+ⅈx2y1-x1y2x22+y22 z1 z2 z1 z2 z2 z2 x1 x2 y1 y2 x2 y1 x1 y2 x2 2 y2 2
Definition 5: Magnitude of a Complex Number
|z1|≡z1z1¯=x12+y12 z1 z1 z1 x1 2 y1 2

Polar Representation

In addition to the Cartesian representation z=x+ⅈy z x y one also has the polar form

z=|z|cosθ+ⅈsinθ z z θ θ (1)
where θ=arctanyx θ y x .

This form is especially convenient with regards to multiplication. More precisely,

z1z2=|z1||z2|cosθ1cosθ2-sinθ1sinθ2+ⅈcosθ1sinθ2+sinθ1cosθ2=|z1||z2|cosθ1+θ2+ⅈsinθ1+θ2 z1 z2 z1 z2 θ1 θ2 θ1 θ2 θ1 θ2 θ1 θ2 z1 z2 θ1 θ2 θ1 θ2 (2)
As a result:
zn=|z|ncosnθ+ⅈsinnθ z n z n n θ n θ (3)

Complex Vectors and Matrices

A complex vector (matrix) is simply a vector (matrix) of complex numbers. Vector and matrix addition proceed, as in the real case, from elementwise addition. The dot or inner product of two complex vectors requires, however, a little modification. This is evident when we try to use the old notion to define the length of a complex vector. To wit, note that if: z=1+ⅈ1-ⅈ z 1 1 then zTz=1+ⅈ2+1-ⅈ2=1+2ⅈ-1+1-2ⅈ-1=0 z z 1 2 1 2 1 2 1 1 2 1 0 Now length should measure the distance from a point to the origin and should only be zero for the zero vector. The fix, as you have probably guessed, is to sum the squares of the magnitudes of the components of zz. This is accomplished by simply conjugating one of the vectors. Namely, we define the length of a complex vector via:

z=z¯Tz z z z (4)
In the example above this produces |1+ⅈ|2+|1-ⅈ|2=4=2 1 2 1 2 4 2 As each real number is the conjugate of itself, this new definition subsumes its real counterpart.

The notion of magnitude also gives us a way to define limits and hence will permit us to introduce complex calculus. We say that the sequence of complex numbers, {zn|n=12…} n 1 2 … zn , converges to the complex number z0z0 and write zn→z0 zn z0 or z0=limn→∞zn z0 n zn when, presented with any ε>0 ε 0 one can produce an integer NN for which |zn-z0|<ε zn z0 ε when n≥N n N . As an example, we note that ⅈ2n→0 2 n 0 .

Examples

Example 1

As an example both of a complex matrix and some of the rules of complex arithmetic, let us examine the following matrix:

F=11111ⅈ-1-ⅈ1-11-11-ⅈ-1ⅈ F 1 1 1 1 1 -1 1 -1 1 -1 1 -1 (5)

Let us attempt to find FF¯ F F . One option is simply to multiply the two matrices by brute force, but this particular matrix has some remarkable qualities that make the job significantly easier. Specifically, we can note that every element not on the diagonal of the resultant matrix is equal to 0. Furthermore, each element on the diagonal is 4. Hence, we quickly arrive at the matrix

FF¯=4000040000400004=4ⅈ F F 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4 4 (6)
This final observation, that this matrix multiplied by its transpose yields a constant times the identity matrix, is indeed remarkable. This particular matrix is an example of a Fourier matrix, and enjoys a number of interesting properties. The property outlined above can be generalized for any FnFn, where FF refers to a Fourier matrix with nn rows and columns:
FnFn¯=nI Fn Fn n I (7)

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