Matrices /Determinants ( 2 x2 )
|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
* 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)
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*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
* 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
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
* 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