Double-Dot Product between 2 Dyads/Dyadics

1. Calculating Double Dot Product between 2 Dyads/Dyadics is same as Double Dot Product between 2 Matrices. For calculating Double Dot Product between 2 Dyads/Dyadics the Order of the both Dyads/Dyadics must be same. The result of Double Dot Product between 2 Dyads/Dyadics is a Scalar Value.
2. Given a Dyad \(\overleftrightarrow{AB}\) formed of Vectors \(\vec{A}\) and \(\vec{B}\) and another Dyad \(\overleftrightarrow{CD}\) formed of Vectors \(\vec{C}\) and \(\vec{D}\), the following relations hold true
   
   \(\overleftrightarrow{AB} : \overleftrightarrow{CD}= \overleftrightarrow{AB}^T : \overleftrightarrow{CD}^T=(\vec{A} \cdot \vec{C})(\vec{B} \cdot \vec{D})\)   ...(For Normal Calculation of Double Dot Product)
   
   \(\overleftrightarrow{AB} : \overleftrightarrow{CD}^T= \overleftrightarrow{AB}^T : \overleftrightarrow{CD}=(\vec{A} \cdot \vec{D})(\vec{B} \cdot \vec{C})\)   ...(For Transpose Calculation of Double Dot Product)
3. Given a Dyadic Polynomial Matrix \(\overleftrightarrow{AB}\) formed by addition of Dyads \(\overleftrightarrow{A_1B_1}\), \(\overleftrightarrow{A_2B_2}\), ... , \(\overleftrightarrow{A_KB_K}\) and another Dyadic Polynomial Matrix \(\overleftrightarrow{CD}\) formed by addition of Dyads \(\overleftrightarrow{C_1D_1}\), \(\overleftrightarrow{C_2D_2}\), ... , \(\overleftrightarrow{C_ND_N}\), the following relations hold true
   
   \(\overleftrightarrow{AB} : \overleftrightarrow{CD}= \overleftrightarrow{AB}^T : \overleftrightarrow{CD}^T=\sum_{i=1}^{K}\sum_{j=1}^{N}(\vec{A_i} \cdot \vec{C_j})(\vec{B_i} \cdot \vec{D_j})\)   ...(For Normal Calculation of Double Dot Product)
   
   \(\overleftrightarrow{AB} : \overleftrightarrow{CD}^T= \overleftrightarrow{AB}^T : \overleftrightarrow{CD}=\sum_{i=1}^{K}\sum_{j=1}^{N}(\vec{A_i} \cdot \vec{D_j})(\vec{B_i} \cdot \vec{C_j})\)   ...(For Transpose Calculation of Double Dot Product)
4. Given Identity Dyadic \(\overleftrightarrow{I}\) and any other Dyadic \(\overleftrightarrow{AB}\) , the following relation holds true
   
   \(\overleftrightarrow{I}:\overleftrightarrow{AB}= \overleftrightarrow{AB}:\overleftrightarrow{I}=Trace(\overleftrightarrow{AB})\)
5. Given 2 3-Dimensional Vectors \(\vec{A}\) and \(\vec{B}\) , the following relation related to Identity Dyadic holds true
   
   \((\overleftrightarrow{I} \times \vec{A}) : (\overleftrightarrow{I} \times \vec{B})= (\vec{A} \times \overleftrightarrow{I}) : (\vec{B} \times \overleftrightarrow{I} )=(\overleftrightarrow{I} \times \vec{B}) : (\overleftrightarrow{I} \times \vec{A})= (\vec{B} \times \overleftrightarrow{I}) : (\vec{A} \times \overleftrightarrow{I} )=[\vec{A}_{\times}]:[\vec{B}_{\times}] =[\vec{B}_{\times}]:[\vec{A}_{\times}]= 2(\vec{A} \cdot \vec{B})\)
   
   In above equation \(\vec{A}_{\times}\) and \(\vec{B}_{\times}\) are Skew-Symmetric Matrices corresponding to Vectors \(\vec{A}\) and \(\vec{B}\) respectively.