Matrices /Determinants ( 2 x2 )

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 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 Matrices (  A = 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=b22a12*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