Dual of a Vector/Matrix

1. The Dual Vector \(A_D\) of any Vector (Column Matrix) \(A\) can be found out by Multiplying the Reciprocal of the Dot Product of the Vector with itself with the Vector. That is
   
   \(A_D= {\Large \frac{A}{A\cdot A}}= {\Large \frac{A}{A^TA}}\)   ...(1)
   
   In such cases, the Dual Vector is the Inverse of the Vector as the Dot Product between the Vector and it's Dual gives the Scalar Value 1. That is
   
   \(A \cdot A_D = A_D \cdot A ={A_D}^TA =A^T{A_D}=1\)   ...(2)
   
   If \(A\) is a Complex Vector, it's Dual Vector \(A_D\) can also be found out by Multiplying the Reciprocal of the Dot Product of the Vector with it's Conjugate with the Vector. That is
   
   \(A_D= {\Large \frac{A}{\overline{A} \cdot A}}= {\Large \frac{A}{A^{\dagger}A}}\)   ...(3)
   
   In such cases, the Conjugate of the Dual Vector is the Inverse of the Vector as the Dot Product between the Vector and Conjugate of it's Dual gives the Scalar Value 1. That is
   
   \(A \cdot \overline{A_D} = \overline{A_D} \cdot A ={A_D}^{\dagger}A=1\)   ...(4)
   
   Also, in such cases, the Dual Vector is the Inverse of the Conjugate of the Vector as the Dot Product between the Conjugate of Vector and it's Dual gives the Scalar Value 1. That is
   
   \(\overline{A} \cdot A_D = A_D \cdot \overline{A} =A^{\dagger}A_D=1\)   ...(5)
2. Dual Matrices exists only for Basis Vector Matrices.
3. The Dual Matrix \(A_D\) of any \(M \times N\) Basis Vector Matrix \(A\) can be found out by Multiplying the Matrix \(A\) with its corresponding Inverse Metric Tensor \({(A^TA)}^{-1}\). That is
   
   \(A_D=A {(A^TA)}^{-1}\)   ...(6)
   
   In such cases, Multiplying the Transpose of Dual Matrix with the Matrix (or Transpose of Matrix with the Dual Matrix) gives an Identity Matrix. That is
   
   \({A_D}^TA =A^T{A_D}=I\)   ...(7)
   
   If \(A\) is a Complex Matrix, the Dual Matrix \(A_D\) can be also found out by Multiplying the Matrix \(A\) with its corresponding Inverse Metric Tensor \({(A^{\dagger}A)}^{-1}\). That is
   
   \(A_D=A {(A^\dagger A)}^{-1}\)   ...(8)
   
   In such cases, Multiplying the Conjugate Transpose of Dual Matrix with the Matrix (or Conjugate Transpose of Matrix with the Dual Matrix) gives an Identity Matrix. That is
   
   \({A_D}^\dagger A =A^\dagger {A_D}=I\)   ...(9)
   
   The Order of Identity Matrix is same as the Number of Columns in the Matrix \(A\).
   
   The Dual Matrix \(A_D\) obtained through either formulae (as given in equation (6) and (8)) is also a \(M \times N\) Basis Vector Matrix. The Vectors (Columns) of Dual Matrix \(A_D\) are called the Dual Basis Vectors of the Corresponding Basis Vectors of Matrix \(A\).
4. The Dual Matrix \(A_D\) of any \(N \times N\) Square Basis Vector Matrix \(A\) found using equation (6) above has following properties
   1. The Dual Matrix is same as Transpose of the Inverse of the Matrix ( or Inverse of the Transpose of the Matrix) as follows
      
      \(A_D = (A^{-1})^T = (A^{T})^{-1} = \frac{Cofactor(A)}{|A|}\)   ...(10)
   2. The Inverse of the Dual Matrix is the Transpose of the Matrix as follows
      
      \({A_D}^{-1} = A^T\)   ...(11)
   3. The Transpose of the Dual Matrix is the Inverse of the Matrix as follows
      
      \({A_D}^{T} = A^{-1}\)   ...(12)
5. The Dual Matrix \(A_D\) of any \(N \times N\) Complex Square Basis Vector Matrix \(A\) found using equation (8) above has following properties
   1. The Dual Matrix is same as Conjugate Transpose of the Inverse of the Matrix ( or Inverse of the Conjugate Transpose of the Matrix) as follows
      
      \(A_D = (A^{-1})^\dagger = (A^{\dagger})^{-1} = \frac{Cofactor(A)}{|A|}\)   ...(13)
   2. The Inverse of the Dual Matrix is the Conjugate Transpose of the Matrix as follows
      
      \({A_D}^{-1} = A^\dagger\)   ...(14)
   3. The Conjugate Transpose of the Dual Matrix is the Inverse of the Matrix as follows
      
      \({A_D}^\dagger = A^{-1}\)   ...(15)
6. The Dual Matrix of any Orthogonal or Unitary Matrix is the Matrix itself.
7. Any Vector/Matrix is a Dual of it's Dual i.e., if \(A_D\) is a Dual Vector/Matrix of any Vector/Matrix \(A\), then \(A\) is a Dual Vector/Matrix of the Vector/Matrix \(A_D\).
8. You can use the Dual of Vector/Matrix Calculator to calculate Dual Vectors/Matrices.