Complex Number Manipulation

Let's run through a few elementary manipulations of complex numbers in Matlab:

>> x = 1; 
>> y = 2;
>> z = x + j * y

z =
   1.0000 + 2.0000i

>> 1/z

ans =
   0.2000 - 0.4000i

>> z^2

ans =
  -3.0000 + 4.0000i

>> conj(z)

ans =
   1.0000 - 2.0000i

>> z*conj(z)

ans =
     5

>> abs(z)^2

ans =
    5.0000

>> norm(z)^2

ans =
    5.0000

>> angle(z)

ans =
    1.1071

Now let's do polar form:

>> r = abs(z)

r =
    2.2361

>> theta = angle(z)

theta =
    1.1071

Curiously, $e$ is not defined by default in Matlab (though it is in Octave). It can easily be computed in Matlab as e=exp(1). Below are some examples involving imaginary exponentials:

>> r * exp(j * theta)

ans =
   1.0000 + 2.0000i

>> z

z =
   1.0000 + 2.0000i

>> z/abs(z)

ans =
   0.4472 + 0.8944i

>> exp(j*theta)

ans =
   0.4472 + 0.8944i

>> z/conj(z)

ans =
  -0.6000 + 0.8000i

>> exp(2*j*theta)

ans =
  -0.6000 + 0.8000i

>> imag(log(z/abs(z)))

ans =
    1.1071

>> theta

theta =
    1.1071

>>

Some manipulations involving two complex numbers:

>> x1 = 1;
>> x2 = 2;
>> y1 = 3;
>> y2 = 4;
>> z1 = x1 + j * y1;
>> z2 = x2 + j * y2;
>> z1

z1 =
   1.0000 + 3.0000i

>> z2

z2 =
   2.0000 + 4.0000i

>> z1*z2

ans =
 -10.0000 +10.0000i

>> z1/z2

ans =
   0.7000 + 0.1000i

Another thing to note about Matlab is that the transpose operator ' (for vectors and matrices) conjugates as well as transposes. Use .' to transpose without conjugation:

>>x = [1:4]*j

x =
        0 + 1.0000i   0 + 2.0000i   0 + 3.0000i   0 + 4.0000i

>> x'

ans =
        0 - 1.0000i
        0 - 2.0000i
        0 - 3.0000i
        0 - 4.0000i

>> x.'

ans =
        0 + 1.0000i
        0 + 2.0000i
        0 + 3.0000i
        0 + 4.0000i

>>