.Spot.Base.Types.Complex

Information

Complex number (x + jy), considered as an element of an algebra, and represented by the real 2x2 matrix z:

  z = [x, -y]
      [y,  x]

The following relations hold for

- complex conjugate:   transpose(z)
- addition:            z1 + z2
- multiplication:      z1*z2
- inverse:             transpose(z)/det(z)
- absolute value:      sqrt(det(z))

where det(z) denotes the determinant of matrix z.

Example:

The linear differential equation with constant coefficients for a real scalar current I and voltage V

  L*dI/dt + R*I = V

  (R + j*omega*L)*i = v

  Complex Z =  R*re_unit + omega*L*im_unit (complex impedance)
  Complex i = i1*re_unit + i2*im_unit      ('current phasor')
  Complex v = v1*re_unit + v2*im_unit      ('voltage phasor')

  Z*i = v

Definition

type Complex = Real[2, 2];