2x2 complex matrices

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.