Complex Numbers in Matlab and Octave

Matlab and Octave have the following primitives for complex numbers:

octave:1> help j

j is a built-in constant

 - Built-in Variable: I
 - Built-in Variable: J
 - Built-in Variable: i
 - Built-in Variable: j

A pure imaginary number, defined as `sqrt (-1)'.  The `I' and `J'
forms are true constants, and cannot be modified.  The `i' and `j'
forms are like ordinary variables, and may be used for other
purposes.  However, unlike other variables, they once again assume
their special predefined values if they are cleared *Note Status
of Variables::.

Additional help for built-in functions, operators, and variables
is available in the on-line version of the manual.  Use the command
`help -i <topic>' to search the manual index.

Help and information about Octave is also available on the WWW
at http://www.octave.org and via the help-octave@bevo.che.wisc.edu
mailing list.

octave:2> sqrt(-1)
ans = 0 + 1i

octave:3> help real
real is a built-in mapper function

 - Mapping Function:  real (Z)
     Return the real part of Z.

See also: imag and conj. ...

octave:4> help imag
imag is a built-in mapper function

 - Mapping Function:  imag (Z)
     Return the imaginary part of Z as a real number.

See also: real and conj. ...

octave:5> help conj
conj is a built-in mapper function

 - Mapping Function:  conj (Z)
     Return the complex conjugate of Z, defined as
     `conj (Z)' = X - IY.

See also: real and imag. ...

octave:6> help abs
abs is a built-in mapper function

 - Mapping Function:  abs (Z)
     Compute the magnitude of Z, defined as
     |Z| = `sqrt (x^2 + y^2)'.

     For example,

          abs (3 + 4i)
          => 5
...
octave:7> help angle
angle is a built-in mapper function

 - Mapping Function:  angle (Z)
     See arg.
...
Note how helpful the ``See also'' information is in Octave (and similarly in Matlab).

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 + 2i

>> 1/z

ans =
   0.2 - 0.4i

>> z^2

ans =
  -3 + 4i

>> conj(z)

ans =
   1 - 2i

>> z*conj(z)

ans =
     5

>> abs(z)^2

ans =
    5

>> norm(z)^2

ans =
    5

>> 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 + 2i

>> z

z =
   1 + 2i

>> z/abs(z)

ans =
   0.4472 + 0.8944i

>> exp(j*theta)

ans =
   0.4472 + 0.8944i

>> z/conj(z)

ans =
  -0.6 + 0.8i

>> exp(2*j*theta)

ans =
  -0.6 + 0.8i

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

ans =
    1.1071

>> theta

theta =
    1.1071

>>
Here are 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 + 3i

>> z2

z2 =
   2 + 4i

>> z1*z2

ans =
 -10 +10i

>> z1/z2

ans =
   0.7 + 0.1i

Another thing to note about matlab syntax 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 + 1i   0 + 2i   0 + 3i   0 + 4i

>> x'

ans =
        0 - 1i
        0 - 2i
        0 - 3i
        0 - 4i

>> x.'

ans =
        0 + 1i
        0 + 2i
        0 + 3i
        0 + 4i


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Factoring Polynomials in Matlab
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Solving Linear Equations Using Matrices