Transposition, symmetrization, alternation. Symmetric and antisymmetric tensors (36.30-36.57).

of type $(1, 1)$

of type $(2, 0)$?

$a_{ij}$

$a^{ij}$

$a_{k}^{ij}$

$a_{jk}^{i}$

How many distinct tensors can be obtained by means of the transposition operation from a given tensor $a_{i_{1} \ldots i_{k}}?$

A tensor of type $(0, 3)$ is given by the matrix $\mathbf{A}_{670}$. Write out the matrices of all tensors obtained from it by transposition. Will the answer change if the given tensor has type $(3, 0)?$. $\mathbf{A}_{670} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 5 & 6 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 3 & 4 \\ 7 & 8 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

A tensor $a$ with components $a_{ijk}$ is given by the matrix $\mathbf{A}_{727}$. Write out the matrices of the transposed tensors $b$ and $c$, if $b_{ijk} = a_{jki}, c_{ijk} = a_{ikj}$. $\mathbf{A}_{727} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 3 & 2 \\ 9 & 7 & 8 \\ 4 & 2 & 3 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 7 & 9 & 8 \\ 3 & 1 & 2 \\ 8 & 6 & 7 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 4 & 6 & 5 \\ 6 & 4 & 5 \\ 1 & 9 & 5 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

A tensor $a$ with components $a_{ijkl}$ is given by the matrix $\mathbf{A}_{717}$. Write out the matrices of the transposed tensors $b$ and $c$, if $b_{ijkl} = a_{kjli}; c_{ijkl} = a_{lkji}$. $\mathbf{A}_{717} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 5 & 6 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 3 & 4 \\ 7 & 8 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} 9 & 10 \\ 13 & 14 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 11 & 12 \\ 15 & 16 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

$x^{i} y^{k}$

$x^{(i} y^{k)}$

$x^{[i} y^{k]}$

$x^{i} a_{jk}$

$x^{i} a_{ik}$

$x^{k} a_{i}^{i}$

$x^{(k} a_{i}^{i)}$

$x^{[k} a_{i}^{i]}$

$a_{i}^{i} a_{k}^{k}$

$a^{k}_{[i} a^{i}_{k]}$

$a_{(i}^{i} a_{k)}^{k}$

$\delta_{j}^{i} a_{i}^{k}$

$\delta_{j}^{i} a_{k}^{j}$

$\delta_{j}^{i} a_{i}^{j}$

$\delta_{j}^{i} a_{l}^{k}$

$a_{l}^{k} \delta_{j}^{i}$

$\mathbf{A}_{10}$. $\mathbf{A}_{10} = \left[\begin{smallmatrix} 2 & 5 \\ 1 & 3 \end{smallmatrix}\right]$.

$\mathbf{A}_{77}$. $\mathbf{A}_{77} = \left[\begin{smallmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{smallmatrix}\right]$.

$\mathbf{A}_{24}$. $\mathbf{A}_{24} = \left[\begin{smallmatrix} 1 & -1 \\ -1 & 2 \end{smallmatrix}\right]$.

$\mathbf{A}_{232}$. $\mathbf{A}_{232} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 1 & 1 & 1 \end{smallmatrix}\right]$.

$\mathbf{A}_{650}$. $\mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

$\mathbf{A}_{651}$. $\mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

$\mathbf{A}_{720}$. $\mathbf{A}_{720} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 2 & 4 & 6 \\ 6 & 2 & 4 \\ 4 & 6 & 2 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 4 & 6 & 2 \\ 2 & 4 & 6 \\ 6 & 2 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 6 & 2 & 4 \\ 4 & 6 & 2 \\ 2 & 4 & 6 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

$\mathbf{A}_{694}$. $\mathbf{A}_{694} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -4 & -3 \\ -2 & -1 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} -4 & -3 \\ -2 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 6 \\ 7 & 8 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

$\mathbf{A}_{684}$. $\mathbf{A}_{684} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 19 & 30 \\ 8 & 13 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 17 & 41 \\ 11 & 17 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

$\mathbf{A}_{650}$. $\mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

$\mathbf{A}_{651}$. $\mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

$\mathbf{A}_{720}$. $\mathbf{A}_{720} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 2 & 4 & 6 \\ 6 & 2 & 4 \\ 4 & 6 & 2 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 4 & 6 & 2 \\ 2 & 4 & 6 \\ 6 & 2 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 6 & 2 & 4 \\ 4 & 6 & 2 \\ 2 & 4 & 6 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

$\mathbf{A}_{694}$. $\mathbf{A}_{694} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -4 & -3 \\ -2 & -1 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} -4 & -3 \\ -2 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 6 \\ 7 & 8 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

$\mathbf{A}_{684}$. $\mathbf{A}_{684} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 19 & 30 \\ 8 & 13 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 17 & 41 \\ 11 & 17 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

$\mathbf{A}_{726}$. $\mathbf{A}_{726} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 & 3 \\ 4 & 3 & 2 \\ 9 & 8 & 7 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 4 & 5 & 6 \\ 1 & 5 & 9 \\ 6 & 5 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 7 & 8 & 9 \\ 8 & 7 & 6 \\ 3 & 2 & 1 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

$\mathbf{A}_{721}$. $\mathbf{A}_{721} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 2 & 4 & 6 \\ 6 & 2 & 8 \\ 4 & 6 & 2 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 4 & 6 & 2 \\ 2 & 6 & 4 \\ 8 & 2 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 6 & 8 & 4 \\ 4 & 6 & 2 \\ 2 & 4 & 6 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

$\mathbf{A}_{723}$. $\mathbf{A}_{723} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 0 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & -2 & 0 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 0 & -2 \\ 0 & 0 & 0 \\ 2 & 0 & 0 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 2 & 0 \\ -2 & 0 & 0 \\ 0 & 0 & 0 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

$\mathbf{A}_{725}$. $\mathbf{A}_{725} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 0 & 0 & 0 \\ 1 & -2 & 1 \\ -1 & 2 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -1 & 2 & -1 \\ 0 & 0 & 0 \\ 1 & -2 & 1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & -2 & 1 \\ -1 & 2 & 1 \\ 0 & 0 & 0 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

$\mathbf{A}_{720}$. $\mathbf{A}_{720} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 2 & 4 & 6 \\ 6 & 2 & 4 \\ 4 & 6 & 2 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 4 & 6 & 2 \\ 2 & 4 & 6 \\ 6 & 2 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 6 & 2 & 4 \\ 4 & 6 & 2 \\ 2 & 4 & 6 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

$\mathbf{A}_{650}$. $\mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right]$.

$\mathbf{A}_{722}$. $\mathbf{A}_{722} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 2 & -2 \\ -2 & 0 & 2 \\ 2 & -2 & 0 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{smallmatrix}\right]\end{smallmatrix}\right]$.

$\mathbf{A}_{58}$. $\mathbf{A}_{58} = \left[\begin{smallmatrix} 1 & -2 \\ -2 & 5 \end{smallmatrix}\right]$.

$\mathbf{A}_{207}$. $\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right]$.

Prove that the tensor $\varepsilon_{i_{1}} \ldots i_{n}$ (see problem 35. 21) is skew-symmetric with respect to any pair of indices.

Prove that the tensor $\varepsilon_{i_{1} \ldots i_{n}}$ is skew-symmetric with respect to any subset of the set of indices.

Prove that the tensor $\delta_{j_{1} \ldots j_{k}}^{i_{1} \ldots i_{k}}$ (see problem 35. 20) is skew-symmetric with respect to any pair of upper indices.

Prove that the tensor $\delta_{j_{1} \ldots j_{k}}^{i_{1} \ldots i_{k}}$ is skew-symmetric with respect to any subset of the set of upper indices.

Prove statement 4) for lower indices.

$\delta_{j_{1} \ldots j_{k}}^{i_{1} \ldots i_{k}} a^{j_{1} \ldots j_{k}} = a^{\left[i_{1} \ldots i_{k}\right]}$;

$\delta_{j_{1} \ldots j_{k}}^{i_{1} \ldots i_{k}} b_{i_{1} \ldots i_{k}} = b_{\left[j_{1} \ldots j_{k}\right]}$.

$\delta_{j}^{i} \delta_{l}^{j} \delta_{k}^{l} \delta_{i}^{k}$

$\delta_{j}^{i} \delta_{k}^{j} \delta_{l}^{k} \delta_{m}^{l}$

$\delta_{[i}^{i} \delta_{j}^{j} \delta_{k]}^{k}$

$\delta_{(i}^{i} \delta_{j}^{j} \delta_{k)}^{k}$

$\delta_{j}^{i} \delta_{i}^{j} \delta_{l}^{k} a_{k}^{l}$.

Let a tensor be symmetric with respect to some pair of indices. Prove that the symmetrization operation with respect to these indices does not change the tensor, while the alternation operation gives the zero tensor.

Let a tensor be antisymmetric with respect to some pair of indices. Prove that the symmetrization operation with respect to these indices gives the zero tensor, while the alternation operation does not change the tensor.

Prove that for a tensor symmetric with respect to the first two indices, the identity holds

Prove that for a tensor antisymmetric with respect to the first two indices, the identity holds

A tensor of type $(0, 3)$ is symmetric with respect to the first two and symmetric with respect to the last two indices. Prove that it is also symmetric with respect to the first and third indices.

A tensor of type $(0, 3)$ is antisymmetric with respect to the first two and antisymmetric with respect to the last two indices. Prove that it is also antisymmetric with respect to the first and third indices.

Let a tensor of type $(0, 3)$ be symmetric with respect to the first two indices and antisymmetric with respect to the last two indices. Prove that it is zero.

Give an example of a tensor of type $(0, 3)$ for which $a_{\left[ijk\right]} = 0$, but which is not symmetric with respect to the three indices.

Give an example of a tensor of type $(0, 3)$ for which $a_{(ijk)} = 0$, but which is not antisymmetric with respect to the three indices.

Prove that for a nonzero tensor $a$ of type $(0, 3)$ it is possible for the equalities $a_{(ijk)} = 0$ and $a_{\left[ijk\right]} = 0$ to hold simultaneously.

$\mathbf{A}_{49}$. $\mathbf{A}_{49} = \left[\begin{smallmatrix} 4 & -3 \\ 12 & -8 \end{smallmatrix}\right]$.

$\mathbf{A}_{16}$. $\mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right]$.

$\mathbf{A}_{234}$. $\mathbf{A}_{234} = \left[\begin{smallmatrix} 0 & 2 & 2 \\ 0 & -1 & -2 \\ 0 & 0 & -2 \end{smallmatrix}\right]$.

$\delta_{kl}^{ij}$;

$\delta_{j_{1} \ldots j_{k}}^{i_{1} \ldots i_{k}}$ (see problems $35. 18, 35. 20$).

Let a symmetric tensor $a$ of type $(0, 2)$ have, in some basis, a matrix of rank $r$. Prove that there exist $r$ linearly independent covectors $\mathrm{f}_{1}, \ldots , \mathrm{f}_{r}$, such that $a = \sum_{\alpha = 1}^{r} \mathrm{f}_{\alpha } \otimes \mathrm{f}_{\alpha }$.

State and prove the converse statement.

A quadratic function $\varphi$ on $\mathcal{L}_{2}$ is given by the matrix $\mathbf{A}_{56}$. Represent it as a sum of the squares of two linear functions. Is this representation unique? $\mathbf{A}_{56} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 3 \end{smallmatrix}\right]$.