2x2 complex matrices

 part-1 part-2 part-3 part-4 Matrix : A pure real pure imaginary mixed:real +imaginary mixed:imaginary +real (a1) + i (a2) (b1) + i (b2) (a1) (b1) (a2) (b2) (a1) (b1) (a2) (b2) (c1) + i (c2) (d1) + i (d2) (c1) (d1) (c2) (d2) (c2) (d2) (c1) (d1) det: + i tr: + i det: tr: det: tr: det: tr: det: tr: tr/2: + i (tr/2)2-det: + i tr/2: (tr/2)2-det: tr/2: (tr/2)2-det: tr/2: (tr/2)2-det: tr/2: (tr/2)2-det: λ1a: + i λ1b: + i λ1a: λ1a: λ1a: λ1a: λ1b: + i λ1b:+ i λ1b:+ i λ1b:+ i λ1b:+ i λ1b: + i λ1b: + i λ1b: + i λ1b: + i λ1b: + i λ1b: + i λ1:+ i λ1:+ i λ1:+ i λ1:+ i λ1: + i λ1:+ i λ1:+ i λ1:+ i λ1:+ i λ1: + i (y/x)1a: (y/x)1a: (y/x)1a: (y/x)1a: λ1: + i (y/x)1b: +i (y/x)1b: +i (y/x)1b: +i (y/x)1b: +i λ1: + i (y/x)1b: +i (y/x)1b: +i (y/x)1b: +i (y/x)1b: +i (y/x)1a: + i (y/x): +i (y/x): +i (y/x): +i (y/x): +i (y/x)1b: + i (y/x): +i (y/x): +i (y/x): +i (y/x): +i (y/x)1b: + i (y/x)1b: + i (y/x)1b: + i (y/x): + i (y/x): + i (y/x): + i (y/x): + i

 * we take a complex 2x2 matrix, [a1+ia2      b1+ib2                                                      c1+ic2      d1+id2 ]     The determinant is | a1  b1| - |a2  b2|      + i |a1 b1|  +i |a2  b2|                                                                                                                         |  c1 d1|    |c2  d2|           |c2 d2|       |c1  d1| . In the real part, there are 2 sub parts which are subtracted from each other and in the imaginary parts, there are 2 sub parts which are added. In the real sub-part, the matrices are pure i.e. either they are from real (a1,b1,c1,d1) or from imaginary component (a2,b2,c2,d2). In the imaginary sub-part, the matrices are mixed i.e 2 from real  (a1,b1; or c1,d1) + 2 from imaginary  (a2,b2; or c2,d2) components. *if x+iy is a complex number, then √( x+iy) = ( x2+y2)1/4 * [cos [(atan(x/y))/2] +i [sin [(atan(x/y))/2] as per De Moivre Theorem.