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Course by: Steven J. Cox. E-mail the author

# Complex Numbers, Vectors and Matrices

Module by: Steven J. Cox. E-mail the author

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

z1+z2x1+x2+i(y1+y2) z1 z2 x1 x2 y1 y2
Definition 2: Complex Multiplication
z1z2(x1+iy1)(x2+iy2)=x1x2y1y2+i(x1y2+x2y1) z1 z2 x1 y1 x2 y2 x1 x2 y1 y2 x1 y2 x2 y1
Definition 3: Complex Conjugation
z1¯x1iy1 z1 x1 y1
Definition 4: Complex Division
z1z2z1z2z2¯z2¯=x1x2+y1y2+i(x2y1x1y2)x22+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|(cosθ1cosθ2sinθ1sinθ2+i(cosθ1sinθ2+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|n(cosnθ+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+i 1i ) z 1 1 then zTz=1+i2+1i2=1+2i1+12i1=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+|1i|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 znz0 zn z0 or z0=limit  nzn z0 n zn when, presented with any ε>0 ε 0 one can produce an integer NN for which |znz0|<ε zn z0 ε when nN n N . As an example, we note that i2n0 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=( 1111 1i-1i 1-11-1 1i-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¯=( 4000 0400 0040 0004 )=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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