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 = Vis mapped by Fourier-transform to the algebraic equation
(R + j*omega*L)*i = vwith complex i and v, if for simplicity only one frequency term is assumed. Using
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')
the equation finally becomes
Z*i = v
type Complex = Real[2, 2];