Introduction to Dyads and Dyadics Algebra

1. Dyads are Special Type of Square Matrices of Rank 1 obtained by Calculating Tensor Product of 2 Vectors that have Same Number of Elements (i.e are of Same Dimension). Tensor Product of 2 \(N\)-Dimensional Vectors gives rise to a Dyad of Order \(N\).
2. The following demonstrates calculation of Tensor Product between 2 \(N\)-Dimensional Vectors \(\vec{A}\) and \(\vec{B}\) resulting in a Dyad \(\overleftrightarrow{AB}\)
   
   \(\vec{A}=\begin{bmatrix}a_{1} \\ a_{2} \\ \vdots \\ a_n\end{bmatrix}\hspace{.5cm}\vec{B}=\begin{bmatrix}b_{1} \\ b_{2} \\ \vdots \\ b_n\end{bmatrix}\)
   
   \(\overleftrightarrow{AB}= \vec{A} \otimes_{T} \vec{B} = \begin{bmatrix}a_{1} \\ a_{2} \\ \vdots \\ a_n\end{bmatrix} \otimes \begin{bmatrix}b_{1} & b_{2} & \cdots & b_n\end{bmatrix} = \begin{bmatrix}a_1 \cdot b_1 & a_1 \cdot b_2 & \cdots & a_1 \cdot b_n\\a_2 \cdot b_1 & a_2 \cdot b_2 & \cdots & a_2 \cdot b_n \\ \vdots & \vdots & \ddots & \vdots\\a_n \cdot b_1 & a_n \cdot b_2 & \cdots & a_n \cdot b_n \end{bmatrix}\)
3. Following example calculates the Tensor Product of 2 3-Dimensional Vectors \(\vec{A}\) and \(\vec{B}\) resulting in a Dyad \(\overleftrightarrow{AB}\)
   
   \(\vec{A}=\begin{bmatrix}2 \\ -3 \\ -9\end{bmatrix}\hspace{.5cm}\vec{B}=\begin{bmatrix}7 \\-1 \\ 4 \end{bmatrix}\)
   
   \(\overleftrightarrow{AB}= \vec{A} \otimes_{T} \vec{B}=\begin{bmatrix}2 \\ -3 \\ -9\end{bmatrix}\otimes\begin{bmatrix}7 & -1 & 4 \end{bmatrix}=\begin{bmatrix}14 & -2 & 8 \\ -21 & 3 & -12 \\ -63 & 9 & -36\end{bmatrix}\)
   
   
4. Dyadics or Dyadic Polynomials are obtained by Adding 2 or More Dyads. Dyadic Polynomial Matrices obtained by adding Dyads of Order \(N\) can have a Rank of 1 to \(N\).
5. An \(N^{th}\)-Order Dyadic of Rank \(N\) is called a Complete Dyadic. Complete Dyadic Polynomial Matrices of Order \(N\) can be obtained by Adding \(N\) or More Dyads of \(N^{th}\)-Order.