MTH3008 Weekly Problems 7

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7.1. Outer Products of Given Tensors

Question

Compute the outer products of the following pairs of tensors.

1. Let $M=[M_{ij}]=\left(\begin{array}{cc}0& 1\\ 1& -1\end{array}\right)$ and $N=[N_{ij}]=\left(\begin{array}{cc}0& 2\\ 1& 3\end{array}\right)$. Compute the outer product $M\otimes N$.
2. Let $P=[P_i]=\left(\begin{array}{cc}0& 1\end{array}\right)$ and $Q=[Q_{ijkl}]$ be the rank $4$ tensor with components $[Q_{ijkl}]=\left(\begin{array}{cccc}{Q}_{1111}& Q_{1112}& Q_{1211}& Q_{1212}\\ Q_{1121}& Q_{1122}& Q_{1221}& Q_{1222}\\ Q_{2111}& Q_{2112}& Q_{2211}& Q_{2212}\\ Q_{2121}& Q_{2122}& Q_{2221}& Q_{2222}\end{array}\right)=\left(\begin{array}{cccc}0& 1& 2& 0\\ 1& 1& -1& -1\\ 0& 1& 1& 0\\ 1& 0& -1& 3\end{array}\right).$ Compute the outer product $P\otimes Q$.
3. Let $R=[R_i]=\left(\begin{array}{ccc}1& -1& 0\end{array}\right)$ and $S=[S_{ij}]=\left(\begin{array}{ccc}1& 2& 4\\ -1& 0& -7\\ 0& 1& 0\end{array}\right)$. Compute the outer product $R\otimes S$.

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7.2. Rank of a Tensor Built from Contraction

Question

Let $T_{ij}$ and $R_{ijk\ell }$ be tensors of rank $2$ and $4$, respectively. Show that the tensor with components $T_{km}{R}_{kpqm}$ is a tensor of rank $2$.

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7.3. Inner Products with a Rank-2 Tensor and a Vector

Question

Given the rank $2$ tensor $R_{ik}$ with matrix representation $[R_{ik}]=\left(\begin{array}{ccc}1& 0& 2\\ 3& 2& 1\\ 1& 3& 4\end{array}\right),$ and the vector $A=i_1+2i_2-i_3$, compute the inner products:

1. $R_{ik}{A}_i$,
2. $R_{ik}{A}_k$.

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7.4. Inner Product with Two Vectors

Question

Let $T_{ik}$ and $A$ be as in Problem 7.3, and let $B=4i_1+5i_2+3i_3.$ Compute the scalar inner product $T_{ik}{A}_i{B}_k.$

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7.5. Contractions of Dimension-Two Tensors

Question

For each of the following tensors (in dimension $2$), list all possible distinct contractions.

1. Let $A_{ij}$ be the rank $2$ tensor $[A_{ij}]=\left(\begin{array}{cc}1& -1\\ 4& 0\end{array}\right).$ Find all possible contractions of $A_{ij}$.
2. Let $B_{ijk\ell }$ be the rank $4$ tensor with components $[B_{ijk\ell }]=\left(\begin{array}{cccc}{B}_{1111}& B_{1112}& B_{1211}& B_{1212}\\ B_{1121}& B_{1122}& B_{1221}& B_{1222}\\ B_{2111}& B_{2112}& B_{2211}& B_{2212}\\ B_{2121}& B_{2122}& B_{2221}& B_{2222}\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 7\\ 4& 0& 2& 11\\ 2& 2& -1& -3\\ 1& 128& 2& 0\end{array}\right).$ Find all possible distinct contractions of $B_{ijk\ell }$.

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