Matrices /Determinants ( 2 x2 )

 

ii
Matrix A a11 a12
  a21 a22
eigen value 1:                                                       eigen value 2 : Det |A| Trace A
Eigen vectors are (x1,x2) 2 sets x1  
      *x1(eigen vector) =*x2(eigen vector) *x1(eigen vector) =*x2(eigen v) x2  
      *x1(eigen vector) =*x2(eigen vector) *x1(eigen vector) =*x2(eigen v)        
                ** Put a value of x1->    x2->  or 

                ** Put a value of x1->    x2->  or 

       
Det |A| is the area of parallelogram ABCD  with co-ordinates of   A (a1,a2) (0)   (0)  
                                                                                                         B (b1,b2) (a11)   (a12)  
                                                                                                         C (c1,c2) (a11+a21)   (a12+a22)  
                                                                                                         D (d1,d2) (a21)   (a22)  
which is twice triangle ABC =2*| a1  a2 1 |

                                                    | b1  b2 1 | =a11*a22-a12*a21=det |A|

                                                    | c1  c2 1 |

       
         
Matrix B b11 b12
  b21 b22
eigen value 1:                         eigen value 2 : Det |B| Trace B
          
AB ab11 ab12
  ab21 ab22
  Det |AB| Trace AB
       
BA ba11 ba12
  ba21 ba22
  Det |BA| Trace BA
       
AAt aat11 aat12
  aat21 aat22
any square matrix multiplied by its transpose produces a symmetric matrix Det |AAt | Trace AAt
         
BBt bbt11 bbt12
  bbt21 bbt22
  Det |BBt | Trace BBt
         
AtA ata11 ata12
  ata21 ata22
  Det |AtA| Trace AtA
         
BtB btb11 btb12
  btb21 btb22
  Det |BtB| Trace BtB
         
Finding pair of 2x2 matrices that anti-commute        
(Y) = (m) *(X) + (c) a11 a12
            Y =        m X  + c a21 a22
         
  b11 b12
b21 b22
         
Finding pair of 2x2 matrices that  invert anti-commute with diagonal elements zero. a11 a12
             X(a11)Y(b11) = X1(a22)Y1(b22)=constant(-a21*b12) Equn. of rectangular hyperbola a21 a22
         
  b11 b12
b21 b22
         
Finding pair of 2x2 matrices that  commute a11 a12
 Y (b21) =P (a21/a12) X(b12) a21 a22
         
  b11 b12
b21 b22
         
Case when a12=-a21   and b12=-b21     a12
  a21    
      b12
b21    
         
Constructing Orthogonal Matrix :- pre -orthogonal matrix set-1 (po11 < or = 1 ) po11 po12
po21 po22
     
co11,co12,co21 value to be put +1 or -1 only. co11 co12
  co21 co22
orthogonal matrix (set 1 ) o11 o12
  o21 o22
  Det:    
Angle θ in Degree      
Period of the Matrix (n-1)      
Constructing Orthogonal Matrix :- pre -orthogonal matrix set-2 (tpo11 < or = 1 ) tpo11 tpo12
tpo21 tpo22
         
co11,co12,co21 value to be put +1 or -1 only. tco11 tco12
  tco21 tco22
orthogonal matrix (set 2 ) to11 to12
  to21 to22
  Det:    
Angle θ in Degree      
Period of the Matrix (n-1)      
         
Orthogonal Set 1 ( from angle) Angle in Degree      
cos A      sinA oo11 oo12
-sinA      cosA oo21 oo22
  Det:    
Orthogonal Set 2 ( from angle) Angle in Degree      
sin A       cos A too11 too12
-cos A     sin A too21 too22
  Det:    
Orthogonal Set 3 ( from angle) Angle in Degree      
sin A       cos A ttoo11 ttoo12
cos A     -sin A ttoo21 ttoo22
  Det:    
         
Constructing Distance Preserving Matrices ds11 ds12
  ds21 ds22
         
  bds11 bds12
  bds21 bds22
       
Constructing Symmetric matrix from a Square Matrix A sym11 sym12
  sym21 sym22
Constructing Skew-Symmetric matrix from a Square Matrix A ssym11 ssym12
  ssym21 ssym22
Constructing Symmetric matrix from a Square Matrix B bsym11 bsym12
  bsym21 bsym22
Constructing Skew-Symmetric matrix from a Square Matrix B bssym11 bssym12
  bssym21 bssym22
         
Constructing Involutory MatricesA = A-1       ) ina11 ina12
ina22 should be between -1 to +1 ina21 ina22
Constructing Involutory Matrices (  B = B-1       ) bina11 bina12
bina22 should be between -1 to +1 bina21 bina22
       
Constructing Self-Adjoint Matrices (Set-1) ad11 ad12
  ad21 ad22
Constructing Self-Adjoint Matrices (Set-2) bd11 bd12
  bd21 bd22
       
Matrix whose adjoint is equal to its inverse (set-1) adi11 adi12
  adi21 adi22
Matrix whose adjoint is equal to its inverse (set-2) bdi11 bdi12
  bdi21 bdi22
       
Transformation of Rectilinear Numbers to Polar Numbers in 2-D x    
  y    

Find

r    
  θ in radian θ in degree
         
Transformation of Polar Numbers to Rectilinear Numbers in 2-D r(any no.)    
  θ in radian    

Find

x θ in degree
  y    
         
         
       
Construct a Singular Matrix from Matrix A sa11 sa12
  sa21 sa22
Det & Trace det trace
         
         
         

 

Finding product of 2x2 matrices that anti commute :

Let Matrix A be | a11  a12 |   and       Matrix  B  be     | b11  b12 |

                           | a21  a22 |                                          | b21  b22 |

Then AB  is   | a11*b11 + a12*b21      a11*b12 + a12*b22  |    and  BA  is  |  b11*a11 + b12*a21         b11*a12 + b12*a22  |

                      | a21*b11 + a22*b21      a21*b12 + a22*b22  |                        |  b21*a11 + a21*b22         a12*b21 + b22*a22  |

To anti commute ->  a11*b11 + a12*b21 = -b11*a11 - a21*b12  or    a12*b21 + a21*b12 = -2*a11*b11 .......(1)

                                  a21*b12 +a22*b22 = - a12*b21 - a22*b22  or    a12*b21 + a21*b12 = -2*a22*b22 .......(2)

 or                                                                                                                              a11*b11 =      a22*b22........(3)

and                            a21*b11 + a22*b21 = -a11*b21 - a21*b22  or    a21(b11 + b22 ) + b21(a11 + a22 ) = 0 ......... (4)

                                  a11*b12 + a12*b22 = - a12*b11 - a22*b12 or   a12(b11 + b22 )  +b12(a11 + a22)   = 0 ......... (5)

               (4) + (5)                                                                                   (b11 + b22 )(a12 + a21) +(a11 + a22)(b12 + b21) = 0 ........ (6)

which implies that            (a11 + a22) =   0

                                         (b11 + b22 ) = 0   which satisfies conditions of Equation (3) & hence separate fulfillment of Equation (3)  is not required. However Equation (1) to be satisfied.

or     for invert anti-commute

                                 (a12 + a21) = 0

                                 (b12 + b21) =0   But these 2 equations require separate fulfillment of Equation (3) which becomes a21*b12= - a11*b11   = -a22*b22

*If we multiply two 2x2 matrices with each having trace zero, then in general , we get 2 right hand diagonal values being anti-commutative and diagonal values reversed which is the adjoint matrix.  

* If we multiply two 2x2 matrices with each having sum of RHS diagonals zero, then in general , we get 2 right hand diagonal values being reversed & anti-commutative and diagonal values unchanged.                                                  

* If we have a condition that (a11/a22) = (d22/d11)= 1 instead of -1, then RHS diagonals remain the same, and diagonals are reversed.

* If we have a condition that (a11/a22) = (d22/d11)= 0, then RHS diagonals have 1 element zero with its position reversed and diagonals reversed.

* If we have a condition that (a11/a22) = (d22/d11)= in between -1 & 1 except -1,0,1, diagonals are only reversed. Other 2 elements have no similarity.

or     for commuting matrices

a11=a22  and b11=b22

a12*b21+ (-a21)*b12=0  or b21=(a21/a12)*b12

* If A is a symmetric Matrix and B is also a symmetric matrix, then AB,BA, AB+BA are all symmetric matrices. AB-BA is an anti-symmetric matrix.

* If A is anti-symmetric, B is anti-symmetric. then AB=BA=Diagonal (scalar matrices)

* If A is symmetric , B is anti-symmetric, then AB,BA both trace and off diagonal trace are zero.[ AB + BA ] has diagonal elements zero each & off diagonal trace zero. [AB-BA] has off diagonal elements each zero and trace zero.

Construction of orthogonal matrix: All elements are less than1 & greater than -1, limiting value being +1 and -1 respectively. If one element is  1 or -1, there will be another element with zero value. Following equations to be satisfied. a11*a11+a12*a12=1 .......(1)

                                                                    a21*a21+a22*a22=1........(2)

                                                                    a11*a21+a12*a22=0 or  a11*a21=-a12*a22........(3)

squaring both sides of (3), a2c2=(1-a2)(1-c2)  or a2+c2=1.........(4)

If a12 & a21 are of same sign, a11 & a22 will be of opposite sign and vice versa. Moreover, all the elements will be in the range [-1.+1] & hence can be expressed as Sine A  or Cos A

                                                             

Distance Preserving Matrices :

                                                                    a11*a11+a21*a21=1 .......(1)

                                                                    a22*a22+a12*a12=1........(2)

                                                                    a11*a12+a22*a21=0 or  a11*a12=-a22*a21........(3)

 Solving (3) with (1),(2)                            a22 = +a11  or a22=-a11

If (a11,a22) will be of same sign, (a12,a21) will be of opposite sign and vice versa. Moreover, all the elements will be in the range [-1.+1] & hence can be expressed as Sine A  or Cos A

Diagonalizing Matrix: Any non-singular square matrix M can be diagonalised to D by a suitable matrix S such that S-1MS = D and the diagonal elements are the eigen values of the matrix. The column elements of S are the eigen vectors.If the matrix is real symmetric/ Hermitian, the diagonalizing matrix is orthogonal/unitary. The invariant vectors of a diagonal matrix are unit vectors. If A ,B are two square matrices and A is non-singular, A-1B and BA-1 have the same eigen values.

The vectors | 2 |   and  |5|     are linearly independent since determinant of the matrix is (10*2-5*1) = 20-5=15 which is non-zero.

                    | 1 |         |10|

 

Construction of Matrix whose adjoint is equal to its inverse

a22=(a12*a21+1) /a11;

a11*a22-1=a12*a21

* In General, adjoint of an adjoint, transpose of a transpose, inverse of an inverse are all original matrices. But Reflection of a reflection matrix is also original. Which of the following i.e adjoint, transpose or inverse can be treated as analogous to the reflection matrix ???

Self - Adjoint Matrix : a11=a22

                                       a12=a21=0

                                      determinant = a2

                                             Inverse matrix : a11=a22=1/a ; a21=a12=0

                                     determinant of inverse matrix =1/ a2

Singular Matrices: These are boundary between the matrices whose determinant are positive and the matrices whose determinant are negative.

* Concept of singular matrix is helpful in solving linear Equations

* To construct a Singular Matrix    〉→ find mean of each column, subtract each mean from each value of respective columns, multiply the resulting matrix by its transpose.

* Over a field of q elements, total no. of (2x2) matrices that can be constructed is q4 . No. of non-singular (2x2)matrices over the field is (q2 -1)(q2-q)=q4 - q3 - q2 +q. Hence no. of Singular matrices that can be formed is q4 - ( q4 - q3 - q2 +q ) =q3 + q2 - q

Involutory Matrices :If a matrix is equal to its inverse, it is called an involutory matrix. The following conditions are satisfied in involutory matrices.

a11 =-d11 or Trace of the matrix is zero.

(a11)2+ a12*a21 = 1