COMPLEX SQUARE MATRIX ( 2x2)

 

(to fix a22a,a21a,a12a)   ((to fix a22a,a21a)
(a11) + i (a11a)   (a12) + i (a12a)
(a21) + i (a21a)   (a22) + i (a22a)

Determinant of A     + i
Angle in  (degree)

r=  (value be 1 or less than 1)

Angle in  (degree) ; Angle in  (degree)

r=  (value be 1 or less than 1) ;

 

 

 
The matrix is &      
The matrix is      
The matrix is      
         
Angle in  (degree) sin   send value to
  cos   send value to   
Put any real value for k1& n1 & X      
HERMITIAN MATRIX Det :   SKEW-HERMITIAN Det:
+i   i +i
+i   +i i
A is         
AA (always Hermitian /symmetric) det(  AA) =   AA (always Hermitian/symmetric) det( AA) =
(A11) (A12)+i(A12a)   (nA11) (nA12)+i(nA12a)
(A21)+i(A21a) (A22)   (nA21)+i(nA21a) (nA22)
2nd Matrix: M

Determinant of M     + i
(m11) + i (m11a)   (m12) + i (m12a)
(m21) + i (m21a)   (m22) + i (m22a)
The matrix is &      
The matrix is      
The matrix is      
         
AM Determinant of AM     + i
(am11) + i (am11a)   (am12) + i (am12a)
(am21) + i (am21a)   (am22) + i (am22a)
         
         
         
MA Determinant of MA     + i
(ma11) + i (ma11a)   (ma12) + i (ma12a)
(ma21) + i (ma21a)   (ma22) + i (ma22a)
         
         
         
1,        
PART-1: (real part)    
D1= trace=   D1'= trace=
(a11)   (a12)    
(a21)   (a22)    
λ1=+i   λ2=+i   λ1=+i   λ2=+i
eigenvector x1    eigenvector x1         
vector x2: +i

x2=(λ1-a11)x1/a12

  vector x2: +i

x2=(λ2-a11)x1/a12

       
normalized x1:    normalized x1:         
normalized x2:

(if eigenvector real)

  normalized x2:

(if eigenvector real)

       
(λ1real)2 +(λ2real)2     (λ1real)2 +(λ2real)2  
λ1 probability     λ1 probability  
λ2 probability     λ2 probability  
(a11)2 +(a12)2=          
(a21)2 +(a22)2=          
a11*a21+a12*a22 =          
(a22-a11) /(2a12) o11        
√[(a22-a11)(a22-a11)+(4a12*a21)]/2a12 o12        
Similarity transformation Matrix  T

(T11 or T12)

  Det:        
y111(any value)   y112(any value)        
  y121a      y122a        
Diagonal Matrix   T-1 * D1 * T        
(DM111)   (DM112)        
(DM121)   (DM122)        
             
The matrix D1 is          
The matrix is          
The matrix is          
The matrix is     orthogonal        
The matrix is          
λ1λ2(equals D1)   +i        

Kernel of transformation is a set of vectors that are mapped by a matrix to zero.  Here, the vectors are

x

y

             

 

 

 

 

x

y

        Inverse by another   Method
co-factor:

adjoint :

inverse :

 

1/

D1

 

|a22       -a12|

|-a21      a11|

1/

   

   

PART-2: (real part)    
D2= trace=   D2'= trace=
(a11a)   (a12a)    
(a21a)   (a22a)    
λ1=+i   λ2=+i   λ1=+i   λ2=+i
eigenvector x1    eigenvector x1         
vector x2: +i

x2=(λ1-a11a)x1/a12a

  vector x2: +i

x2=(λ2-a11a)x1/a12a

       
normalized x1:    normalized x1:         
normalized x2:

(if eigenvector real)

  normalized x2:

(if eigenvector real)

       
(λ1real)2 +(λ2real)2     (λ1real)2 +(λ2real)2  
λ1 probability     λ1 probability  
λ2 probability     λ2 probability  
(a11)2 +(a12)2=          
(a21)2 +(a22)2=          
a11*a21+a12*a22 =          
(a22a-a11a) /(2a12a) o21        
√[(a22a-a11a)(a22a-a11a)+(4a12a*a21a)]/2a12a o22        
Similarity transformation Matrix  T

(T21 or T22)

  Det:        
y211(any value)   y212(any value)        
  y221a      y222a        
Diagonal Matrix   T-1 * D2 * T        
(DM211)   (DM212)        
(DM221)   (DM222)        
             
The matrix D2 is          
The matrix is          
The matrix is          
The matrix is     orthogonal        
The matrix is          
λ1λ2(equals D2)   +i        

Kernel of transformation is a set of vectors that are mapped by a matrix to zero.  Here, the vectors are

x

y

    x

y

        Inverse by another   Method
co-factor:

adjoint :

inverse :

 

1/

D2

 

|a22a       -a12a|

|-a21a      a11a|

1/

   

   

PART-3: (imaginary part)    
D3= trace=   D3'= trace=
(a11)   (a12)    
(a21a)   (a22a)    
λ1=+i   λ2=+i   λ1=+i   λ2=+i
eigenvector x1    eigenvector x1         
vector x2: +i

x2=(λ1-a11)x1/a12

  vector x2: +i

x2=(λ2-a11)x1/a12

       
normalized x1:    normalized x1:         
normalized x2:

(if eigenvector real)

  normalized x2:

(if eigenvector real)

       
(λ1real)2 +(λ2real)2     (λ1real)2 +(λ2real)2  
λ1 probability     λ1 probability  
λ2 probability     λ2 probability  
(a11)2 +(a12)2=          
(a21)2 +(a22)2=          
a11*a21+a12*a22 =          
(a22a-a11) /(2a12) o31        
√[(a22a-a11)(a22a-a11)+(4a12*a21a)]/2a12 o32        
Similarity transformation Matrix  T

(T31 or T32)

  Det:        
y311(any value)   y312(any value)        
  y321a      y322a        
Diagonal Matrix   T-1 * D3 * T        
(DM311)   (DM312)        
(DM321)   (DM322)        
             
The matrix D3 is          
The matrix is          
The matrix is          
The matrix is     orthogonal        
The matrix is          
λ1λ2(equals D3)   +i        

Kernel of transformation is a set of vectors that are mapped by a matrix to zero.  Here, the vectors are

x

y

    x

y

        Inverse by another   Method
co-factor:

adjoint :

inverse :

 

1/

D3

 

|a22a       -a12|

|-a21a      a11|

1/

   

   

PART-4:  (imaginary part)    
D4= trace=   D4'= trace=
(a11a)   (a12a)    
(a21)   (a22)    
λ1=+i   λ2=+i   λ1=+i   λ2=+i
eigenvector x1    eigenvector x1         
vector x2: +i

x2=(λ1-a11a)x1/a12a

  vector x2: +i

x2=(λ2-a11a)x1/a12a

       
normalized x1:    normalized x1:         
normalized x2:

(if eigenvector real)

  normalized x2:

(if eigenvector real)

       
(λ1real)2 +(λ2real)2     (λ1real)2 +(λ2real)2  
λ1 probability     λ1 probability  
λ2 probability     λ2 probability  
(a11)2 +(a12)2=          
(a21)2 +(a22)2=          
a11*a21+a12*a22 =          
(a22-a11a) /(2a12a) o41        
√[(a22-a11a)(a22-a11a)+(4a12a*a21)]/2a12a o42        
Similarity transformation Matrix  T

(T41 or T42)

  Det:        
y411(any value)   y412(any value)        
  y421a      y422a        
Diagonal Matrix   T-1 * D4 * T        
(DM411)   (DM412)        
(DM421)   (DM422)        
             
The matrix is          
The matrix is          
The matrix is          
The matrix is     orthogonal        
The matrix is          
λ1λ2(equals D4)   +i        

Kernel of transformation is a set of vectors that are mapped by a matrix to zero.  Here, the vectors are

x

y

    x

y

        Inverse by another   Method
co-factor:

adjoint :

inverse :

 

1/

D4

 

|a22       -a12a|

|-a21      a11a|

1/

   

   

Part- HERMITIAN MATRIX   SKEW- SYMMETRIC COMPLEX MATRIX
Det : +i     Det : +i  
        +        
  +i       +i  
                 
                 
                 
         
Part- HERMITIAN MATRIX   SKEW- SYMMETRIC COMPLEX MATRIX
Det : +i     Det : +i  
        +        
  +i       +i  
                 
                 
                 
         
Part- HERMITIAN MATRIX   SKEW- SYMMETRIC COMPLEX MATRIX
Det : +i     Det : +i  
        +        
  +i       +i  
                 
                 
                 

 

 

Real Part Part-1 a11,a12

a21,a22

  Part-2 a11a,a12a

a21a,a22a

 

Co-efficient of real part (C) Part1- Part2
Imaginary Part Part-3 a11,a12

a21a,a22a

  Part-4 a11a,a12a

a21,a22

Co-efficient of imaginary part:D34 Part3+Part4
B=-trace(D1)=-(a11+a22)  
B2 - 4C  
D3+D4-λ(traceD2)  
λ1   +i
Eigenvector x1 corresponding to λ1 (EV11)   +i
Eigenvector y1 corresponding to λ1 (EV21)   +i
Normalized Eigenvector x1 corresponding to λ1 (nEV11)   +i
Normalized Eigenvector y1 corresponding to λ1 (nEV21)   +i
λ2   +i
Eigenvector x1 corresponding to λ2 (EV12)          +i
Eigenvector y1 corresponding to λ2 (EV22)          +i
Normalized Eigenvector x1 corresponding to λ2  (nEV12)   +i
Normalized Eigenvector y1 corresponding to λ2  (nEV22)   +i
+i  (nEV11)                        +i  (nEV12)          Determinant(U): +i    
+i  (nEV21)                        +i  (nEV22)    
     
+i  (EV11)                        +i  (EV12)          Determinant(U1): +i    
+i  (EV21)                        +i  (EV22)    
λ1 probability  
λ2 probability  
(λ1real)2 +(λ2real)2  
(λ1real)2 +(λ1img)2  
(λ2real)2 +(λ2img)2  
(λ1 / λ2 )=[(λ1real*λ2real+λ1img*λ2img)/(λ2real*λ2real +λ2img*λ2img)] + i[(λ1img*λ2real -λ1real*λ2img)/(λ2real*λ2real +λ2img*λ2img)]   +i
λ1 - λ2   +i
     
Real Part + imaginary part=0    
Real Part Equn: λ2-λ(traceD1)+(D1-D2)=0 or λ2+Bλ+C=0  ;  where  C=D1-D2;    B=-trace(D1)    
 λ1=[-B + √(B2-4C)] / 2 ; --(1) λ2=[-B - √(B2-4C)] / 2 ---(2)                        
Imaginary Part Equn: i[D3+D4-λ(traceD2)]=0 => [D3+D4-λ(traceD2)]=0---(3)    
 Since λ has got a single value as per (3) and 2 values as per (1) & (2), in order to have consistent eigen value,

Condition1 ---trace(D2)should be equal to zero which means a22a=-a11a      

AND

Condition 2 -- Either D3=D4=0 => (a11/a22)=-(a12/a21)*(a21a/a12a)

OR

                    ---D3+D4=0 => a11a(a22-a11) =a12*a21a+a21*a12a;

We classify the 8 variables in a complex matrix into 2 groups out of which 6 can be arbitrarily chosen, 1 from group-1 and 5 from group-2. 

Group-1 : There are 2 variables  a11a,a22a  out of which 1 can be arbitrarily chosen. We choose a11a.

                 a22a=-a11a;

Group-2: There are 6 variables a12,a21,a11,a22,a12a,a21a out of which any 4 can be arbitrarily chosen. Barring a21a,a12a we choose others. 

                  a21a = a11*(-a11a)/a12  and a12a=a22*a11a/a21;

           or,  There are 6 variables a12,a21,a11,a22,a12a,a21a out of which any 5 can be arbitrarily chosen. Barring a21a, we choose others 5 variables.  (a11a is already chosen)

                  a21a=[a11a(a22-a11)-a21*a12a] / a12 ;

   
To extract meaningful eigen values, fill the matrix, submit, then either click submit 1 or submit 2 and then again click submit.    
* In a Hermitian matrix, the diagonal elements are real, whereas in Skew-Hermitian matrix,  diagonal elements are purely imaginary or zero. If A is Skew-Hermitian, both iA and -iA are Hermitian.    

* In a symmetric matrix, if aii=1 & aij =2 or >2 (whole numbers)-----it is called Coxter matrix. If aii=2 and aij=<0(not necessarily whole no.), it is called Schlaffli matrix. In a matrix, aij=0 if i+j-1 > n and aij=i+j-1 otherwise, it is called Henkel matrix.

* If the matrix above have all eigen values real & greater than zero, matrix is positive definite; if all eigen values less than zero, negative definite; if both indefinite and if one or the other is zero, semi-definite positive or negative depending on whether the non-zero eigen value is positive or negative.

* Any real square matrix can be expressed as the sum of a Hermitian and Skew-symmetric complex matrix.

a  b            k1                               [(b+c)/2] +i X                 a-k1                           [(b-c)/2] -iX

        =                                                                           +

c  d           [(b+c)/2] - i X               n1                                 [-(b-c)/2] +iX             d-n1

   

* An arbitraty square matrix can be expressed as the sum of a unique Hermitian and Skew-Hermitian matrix. If A is the square Matrix and        A= B + C, where B is Hermitian and C is Skew-Hermitian, then B = (A + A)/2  and C =(A-A)/2 ;

* Product of any square complex matrix and its transpose conjugate is always Hermitian. If the product is a unit matrix, then the square matrix is called Unitary Matrix. Therefore, in a unitary matrix, the transpose conjugate is also the inverse of the matrix. The sum of a complex square matrix & its transpose conjugate is always Hermitian.

*For a Hermitian matrix to be unitary, the matrix is of the form

                                           x             r(cosA + i*SinA)

                     r(cosA - i*SinA)         -x 

and x2+r2=1 and Trace of the Matrix=0 and x may be zero, positive or negative .

*For a Skew- Hermitian matrix to be unitary, the matrix is of the form

                                           x             r(cosA + i*SinA)

                     -r(cosA - i*SinA)         -x 

and x2+r2=1 and Trace of the Matrix=0 and x is purely imaginary.

*For a Complex matrix to be unitary, the matrix is of the form

 a*   b                       r1(cosB+iSinB)            r(cosA + i*SinA)

                 =

 b*  -a                    r(cosA - i*SinA)         - r1(cosB-iSinB)

or  

a*    b                       r1(cosB+iSinB)            r(cosA + i*SinA)

                 =

 -b*  a                    -r(cosA - i*SinA)          r1(cosB - iSinB)

 

and x2+r2=1 and Trace of the Matrix=0 and x may be zero, positive or negative .

 

A11=(a11*a11+a11a*a11a)+(a12*a12+a12a*a12a) ;

A22=(a21*a21+a21a*a21a)+(a22*a22+a22a*a22a);

A12=(a11*a21+a11a*a21a)+(a12*a22+a12a*a22a);

A12a=(a11a*a21+a12a*a22)-(a11*a21a+a12*a22a);

A21=A12;  A21a=-A12a ;

 

nA11=(a11*a11+a11a*a11a)+(a21*a21+a21a*a21a) ;

nA22=(a12*a12+a12a*a12a)+(a22*a22+a22a*a22a);

nA12=(a11*a12+a11a*a12a)+(a21*a22+a21a*a22a);

nA12a=(a11*a12a+a21*a22a)-(a12*a11a+a22*a21a);

nA21=nA12;  nA21a=-nA12a ;

* Special unitary group of degree n, denoted by SU(n) is a Lie group of nxn Unitary matrices with determinant 1 where the angle is zero and the group operation is that of matrix multiplication. The special Unitary Group is a sub-group of Unitary group U(n) consisting of all n x n Unitary matrices. SU(n) groups find wide applications in Standard Model of Particle Physics, especially SU(2) in electro-weak interaction and SU(3) in Quantum Chromodynamics. SU(2) is isomorphic to the group of quarternions of norm 1.

* Quarternion q is defined as the quotient of 2 vectors. It is of the form a +(bi+cj+dk) where a is the scalar part and that under the bracket is the vector part. The scalar part is always a real munmer, vector part is a purely imaginary no. where b,c,d are real numbers and at least one of b,c,d is non-zero. For every q, there is a conjugate q*. The set H of all quarternions is a vector space over real numbers with dimension 4. Multiplication of quarternions is associative and distributive but not commutative. i*i=j*j=k*k=i*j*k=-1; i*j=k,  j*k=i, k*i=j and j*i=-k, k*j=-i and i*k=-j ; The quarternions can be represented as  2 x2 complex matrices or 4x4 real matrices. For Example, a +(bi+cj+dk) is

a+bi    c+di              a   b    c    d

                     or      -b   a   -d    c

-c+di   a-bi            -c   d     a    -b

                              -d   -c    b     a

* If   (a11)2 +(a12)2=        1

              (a21)2 +(a22)2=        1

and       a11*a21+a12*a22 =0    ,  then the matrix is orthogonal.

 

   
     

 

 (p,q)         (a+b , c+d)           (a,c)    (b,d)    (a,c)  (b,d)                                                                              (p,q)

            =                        =              +          +        +               where p=a+b, q=c+d, r=e+f, s=g+h      where             etc. are the determinants of respective matrices.

  (r,s)        (e+f ,  g+h)           (e,g)     (f,h)      (f,h)   (e,g)                                                                             (r,s)

(p,q)         (a+ib , c+id)         (a,c)    (b,d)     (a,c)   (b,d)                                                                              (p,q)

            =                         =             -          + i       +i            where p=a+ib, q=c+id, r=e+if, s=g+ih  where             etc. are the determinants of respective matrices.

 (r, s)        (e+if ,  g+ih)         (e,g)    (f,h)      (f,h)    (e,g)                                                                               (r,s)

 

(p,q)         [(a+ib)+(a'+ib')] , [(c+id) +(c'+id')]        (a+ib, c+id)     (a'+ib', c'+id')       (a+ib, c+id)      (a'+ib', c'+id') 

            =                                                              =                       +                             +                       +

 (r, s)        [(e+if )+(e'+if')],   [(g+ih) +(g'+ih')]        (e+if, g+ih)       (e'+if', g'+ih')       (e'+if', g'+ih')     (e+if, g+ih)                                                        

 

* If A is a square, non-singular matrix, then it can be diagonalized using a transformation known as Similarity Transformation. If T is a similarity matrix, then           T-1AT=D, a diagonal matrix, the diagonal elements being the eigen values of A. If A is a 2x2 matrix, and

         | p     q|                             | a      b |                      |D11        D12|          |1         0|

T=                           and   A=                        then D=                             =

         | r     s|                               | c      d |                    |D21         D22|          |0         1|

where  D11=[s(ap+br)-q(cp+dr)] /(ps-qr)  ;  D22= [p(ds+cq)-r(aq+bs)]/(ps-rq)  ;  D12=-[bs2 +q(a-d)s -cq2 ] / (ps-rq) ;  D21= -[br2 +p(a-d)r -cp2 ] / (ps-rq) ;

Since in D12=0, denominator is zero and s=q*((d-a) ± √[(d-a)2 + 4bc])/2b                        

Since in D21=0, denominator is zero and r=p*((d-a) ∓ √[(d-a)2 + 4bc])/2b             

        

 then for determining the elements of T, we can arbitrarily assign values to either p or r   and either q or s.  Thus choosing the arbitrary values of any           pairs will do--(p,q),(r,s),(p,s),(q,r) . In our calculation ,we have chosen the arbitrary values of (p,q). Then other two values can be found out.

* If A=  | p   q|

             | r    s|

is a real matrix, then its inverse is |s   -q|

                                                      |-r   p|

divided by the determinant of A.

Similarly, if A is   |p+ip1     q+iq1|

                              |r+ir1       s+is1|

then its inverse is 

 |s+is1         -(q+iq1)|

 |-(r+ir1)         p+ip1|

divided by the determinant of complex A. If Determinant is a +i aa where a =(ps-p1s1+q1r1-qr) and aa=(p1s+ps1-q1r-qr1)

Now the inverse is given by

|[(sa+aa*s1) +i (s1a-s*aa)]/z           [(aa*q1-qa) +i (q*aa-aq1)]/z |

|[(aa*r1-ra) +i (r*aa-ar1)]/z            [(pa+aa*p1) +i (p1a-p*aa)]/z |

where z=a2 +aa2

 

* For the Complex Matrix, to find out the eigenvectors corresponding to eigenvalue λ1+iλ1a, If one eigen vector is x +ix1 where x,x1 are any arbitrary real numbers,

  then the other eigenvector is  y +iy1  where |K   -K1|      |x|         |y|

                                                                                                 =

                                                                      |K1   K |      |x1|       |y1|

where K=a12(λ1-a11)+a12a(λ1a-a11a) / (a12*a12 + a12a*a12a) ;

           K1=a12(λ1a-a11a)-a12a(λ1-a11) / (a12*a12 + a12a*a12a) ;

Similarly for the other eigen value.

 * Any  hermitian   matrix H can be diagonalized by a suitable Unitary matrix U such that UHU =D. Above, we have enumerated how to find U(normalized). First

for First eigenvalue, find pairs of normalized eigenvectors.These pairs constitute the first column of U. for second eigenvalue, find the next pair which constitute  the second column. Thus U is constructed. Among eigenvalue pairs, we have put arbitrary value of first eigenvector and with the relationship, 2nd vector is found. There can be different choices on the matter.