All textbooks/Collection of problems in analytical geometry and linear algebra/14 · Tensors

Section 36

Algebraic operations with tensors

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36.1 · Linear operations, tensor multiplication (36.1-36.20)

36.2 · Collapse (36.21-36.29)

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

Subsection 36.1Linear operations, tensor multiplication (36.1-36.20)[20/75%]

Problem 36.1

Verify the transformation law of the components of the sum of the tensors $a_{jk}^{i}$ and $b_{jk}^{i}$ , starting from the transformation law of the components of the summands.

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Problem 36.2

How is the sum of linear transformations related to the sum of the corresponding tensors?

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Problem 36.3

Let $A_{e}$ be the matrix of a linear transformation $\varphi$ , and let $B_{e}$ be the matrix of a bilinear function $b$ in the basis $e$ . Is the sum $A_{e} + B_{e}$ defined? Will the correspondence assigning to each basis $e$ the sum of matrices $A_{e} + B_{e}$ be a tensor?

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Problem 36.4

(1)(2)(3)(4)

Tensors $a$ and $b$ of the same type have, in a basis $e$ , component matrices $A$ and $B$ . Find the components of the tensors:

(1) $a + b$ ;

(2) $2a + 3b$ ;

(3) $b - 2a$ in the same basis, if:

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(1)

$A = \mathbf{A}_{650}, B = \mathbf{A}_{651}$ . $\mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(2)

$A = \mathbf{A}_{651}, B = \mathbf{A}_{652}$ . $\mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{652} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -4 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 3 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(3)

$A = \mathbf{A}_{652}, B = \mathbf{A}_{653}$ . $\mathbf{A}_{652} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -4 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 3 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{653} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 7 & -4 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 8 \\ 2 & 2 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(4)

$A = \mathbf{A}_{693}, B = \mathbf{A}_{697}$ . $\mathbf{A}_{693} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 0 & 0 \\ 0 & 0 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} -1 & -2 \\ -3 & -4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 0 \\ 0 & 0 \end{smallmatrix}\right]\end{smallmatrix}\right], \mathbf{A}_{697} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 1 \\ 0 & 1 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} 1 & 1 \\ 2 & 2 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 1 \\ 0 & 2 \end{smallmatrix}\right]\end{smallmatrix}\right]$ .

Problem 36.5

(1)(2)(3)

Matrices $A, B, C, D$ of the components of four tensors are given. Determine whether the tensors are linearly dependent, if:

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(1)

$A = \mathbf{A}_{662}, B = \mathbf{A}_{665}, C = \mathbf{A}_{663}, D = \mathbf{A}_{652}$ . $\mathbf{A}_{662} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 11 & 14 \\ 8 & 13 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 17 & 23 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{665} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -6 & 4 \\ 13 & 22 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 23 \\ 17 & 27 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{663} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 0 & 7 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 3 & 5 \\ -1 & 5 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{652} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -4 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 3 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(2)

$A = \mathbf{A}_{650}, B = \mathbf{A}_{651}, C = \mathbf{A}_{652}, D = \mathbf{A}_{653}$ . $\mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{652} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -4 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 3 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{653} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 7 & -4 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 8 \\ 2 & 2 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(3)

$A = \mathbf{A}_{666}, B = \mathbf{A}_{663}, C = \mathbf{A}_{650}, D = \mathbf{A}_{652}$ . $\mathbf{A}_{666} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -10 & -6 \\ -1 & -8 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -15 & -11 \\ 3 & 1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{663} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 0 & 7 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 3 & 5 \\ -1 & 5 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{652} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} -4 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -5 & 3 \\ 5 & 7 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

Problem 36.6

(1)(2)(3)

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(1)

What is the dimension of the vector space $\mathcal{L}$ of tensors of type $(p, q)$ in a two-dimensional space?

(2)

Indicate some basis in $\mathcal{L}$ .

(3)

Indicate another basis in $\mathcal{L}$ .

Problem 36.7

(1)(2)(3)(4)

To a basis $e$ of a two-dimensional vector space there corresponds a basis $\mathbf{e}^{ * }$ in the space of tensors of type:

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(1)

$(0, 1)$

(2)

$(1, 1)$

(3)

(р) $(0, 2)$

(4)

$(1, 2)$

The basis $\mathbf{e}^{*}$ consists of tensors having, in the basis $e$ , one component equal to 1, and the rest equal to 0. How does the basis $\mathbf{e}^{*}$ transform if the basis $e$ is transformed by the transition matrix $S$ ?

Problem 36.8

Verify the transformation law of the components of the tensor $a\otimes b$ , starting from the transformation laws of the components of the factors $a_{jk}^{i}, b_{jk}^{i}$ .

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Problem 36.9

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)

Find the type and matrix of the tensor $a\otimes b$ , if:

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(1)

Type of $a$ : $(1, 0)$ , matrix of $a$ : $\mathbf{c}_{12}$ , type of $b$ : $(1, 0)$ , matrix of $b$ : $\mathbf{c}_{31}$ . $\mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{31} = \left[\begin{smallmatrix} 1 \\ -1 \end{smallmatrix}\right]$ .

(2)

Type of $a$ : $(1, 0)$ , matrix of $a$ : $\mathbf{c}_{12}$ , type of $b$ : $(0, 1)$ , matrix of $b$ : $\mathbf{c}_{31}^T$ . $\mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{31} = \left[\begin{smallmatrix} 1 \\ -1 \end{smallmatrix}\right]$ .

(3)

Type of $a$ : $(1, 0)$ , matrix of $a$ : $\mathbf{c}_{12}^T$ , type of $b$ : $(1, 0)$ , matrix of $b$ : $\mathbf{c}_{31}$ . $\mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{31} = \left[\begin{smallmatrix} 1 \\ -1 \end{smallmatrix}\right]$ .

(4)

Type of $a$ : $(0, 1)$ , matrix of $a$ : $\mathbf{c}_{12}^T$ , type of $b$ : $(0, 1)$ , matrix of $b$ : $\mathbf{c}_{31}^T$ . $\mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{31} = \left[\begin{smallmatrix} 1 \\ -1 \end{smallmatrix}\right]$ .

(5)

Type of $a$ : $(0, 2)$ , matrix of $a$ : $\mathbf{A}_{17}$ , type of $b$ : $(0, 1)$ , matrix of $b$ : $\mathbf{c}_{7}^T$ . $\mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right], \mathbf{c}_{7} = \left[\begin{smallmatrix} 4 \\ -2 \end{smallmatrix}\right]$ .

(6)

Type of $a$ : $(0, 1)$ , matrix of $a$ : $\mathbf{c}_{7}^T$ , type of $b$ : $(0, 2)$ , matrix of $b$ : $\mathbf{A}_{17}$ . $\mathbf{c}_{7} = \left[\begin{smallmatrix} 4 \\ -2 \end{smallmatrix}\right], \mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right]$ .

(7)

Type of $a$ : $(2, 0)$ , matrix of $a$ : $\mathbf{A}_{18}$ , type of $b$ : $(1, 0)$ , matrix of $b$ : $\mathbf{c}_{8}$ . $\mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right], \mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right]$ .

(8)

Type of $a$ : $(1, 1)$ , matrix of $a$ : $\mathbf{A}_{18}$ , type of $b$ : $(1, 0)$ , matrix of $b$ : $\mathbf{c}_{8}$ . $\mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right], \mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right]$ .

(9)

Type of $a$ : $(1, 0)$ , matrix of $a$ : $\mathbf{c}_{8}$ , type of $b$ : $(2, 0)$ , matrix of $b$ : $\mathbf{A}_{18}$ . $\mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right]$ .

(10)

Type of $a$ : $(1, 0)$ , matrix of $a$ : $\mathbf{c}_{8}$ , type of $b$ : $(1, 1)$ , matrix of $b$ : $\mathbf{A}_{18}$ . $\mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right]$ .

(11)

Type of $a$ : $(0, 3)$ , matrix of $a$ : $\mathbf{A}_{650}$ , type of $b$ : $(0, 1)$ , matrix of $b$ : $\mathbf{c}_{25}^T$ . $\mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{c}_{25} = \left[\begin{smallmatrix} 2 \\ 4 \end{smallmatrix}\right]$ .

(12)

Type of $a$ : $(0, 1)$ , matrix of $a$ : $\mathbf{c}_{25}^T$ , type of $b$ : $(0, 3)$ , matrix of $b$ : $\mathbf{A}_{650}$ . $\mathbf{c}_{25} = \left[\begin{smallmatrix} 2 \\ 4 \end{smallmatrix}\right], \mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(13)

Type of $a$ : $(1, 2)$ , matrix of $a$ : $\mathbf{A}_{651}$ , type of $b$ : $(0, 1)$ , matrix of $b$ : $\mathbf{c}_{8}^T$ . $\mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right]$ .

(14)

Type of $a$ : $(0, 1)$ , matrix of $a$ : $\mathbf{c}_{8}^T$ , type of $b$ : $(1, 2)$ , matrix of $b$ : $\mathbf{A}_{651}$ . $\mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(15)

Type of $a$ : $(0, 2)$ , matrix of $a$ : $\mathbf{A}_{17}$ , type of $b$ : $(0, 2)$ , matrix of $b$ : $\mathbf{A}_{18}$ . $\mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right], \mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right]$ .

(16)

Type of $a$ : $(0, 2)$ , matrix of $a$ : $\mathbf{A}_{18}$ , type of $b$ : $(0, 2)$ , matrix of $b$ : $\mathbf{A}_{17}$ . $\mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right], \mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right]$ .

(17)

Type of $a$ : $(1, 1)$ , matrix of $a$ : $\mathbf{A}_{13}$ , type of $b$ : $(1, 1)$ , matrix of $b$ : $\mathbf{A}_{19}$ . $\mathbf{A}_{13} = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right]$ .

(18)

Type of $a$ : $(2, 0)$ , matrix of $a$ : $\mathbf{A}_{13}$ , type of $b$ : $(1, 1)$ , matrix of $b$ : $\mathbf{A}_{19}$ . $\mathbf{A}_{13} = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right]$ .

Problem 36.10

(1)(2)(3)(4)

Write the matrix of the components of the tensor:

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(1)

$a^{i} b^{j}$

(2)

$a_{i} b_{j}$

(3)

$a^{i} b_{j}$

(4)

$a_{i} b^{j}$

as the Kronecker product of the matrices of the components of these tensors.

Problem 36.11

(1)(2)

Let $a, b$ be bivalent tensors with matrices $A, B$ . Of what type must these tensors be so that the matrix of their tensor product is the (right) Kronecker product:

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(1)

$A\otimes B$

(2)

$B\otimes A?$

Problem 36.12

(1)(2)

Linear functions $\mathrm{f}$ and $\mathrm{g}$ are given in a basis $e$ by coordinate rows $\varkappa$ and $\mu$ . Find the matrix of the tensor:

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(1)

$f\otimes g$ ;

(2)

$g\otimes f$ .

What is the geometric meaning of these tensors?

Problem 36.13

A linear function $\mathrm{f}$ is given in a basis $e$ by the coordinate row $\varkappa$ , and a vector $y$ by the column $\eta$ . Find the matrix of the tensor $f\otimes y$ . What is the geometric meaning of this tensor?

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Problem 36.14

(1)(2)

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(1)

Let $x$ be a vector, and $\mathrm{f}$ a covector. Prove that $\mathrm{f} \otimes x = x\otimes \mathrm{f}$ .

(2)

Give an example of tensors $a$ and $b$ for which $a\otimes b \neq b\otimes a$ .

Problem 36.15

(1)(2)(3)(4)(5)(6)(7)(8)

Let $x_{1}, x_{2}, x_{3}$ be vectors, and $\mathrm{f}_{1}, \mathrm{f}_{2}, \mathrm{f}_{3}$ covectors. Which of the expressions below make sense? If the given expression is a tensor, indicate its type:

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(1)

$x_{1} \otimes x_{2} + x_{2} \otimes x_{3}$

(2)

$x_{1} \otimes x_{2} \otimes x_{3} + x_{2} \otimes x_{3}$

(3)

$x_{1} \otimes \mathrm{f}_{1} - 2\mathrm{f}_{1} \otimes x_{1}$

(4)

$x_{1} \otimes \mathrm{f}_{2} + \mathrm{f}_{1} \otimes \mathrm{f}_{2}$

(5)

$x_{1} \otimes \mathrm{f}_{2} + x_{2} \otimes \mathrm{f}_{1}$

(6)

$f_{1} \otimes x_{1} \otimes x_{2} + x_{2} \otimes x_{3} \otimes \mathrm{f}_{2}$

(7)

$x_{1} \otimes x_{2} + x_{3} \otimes x_{3} - x_{1} \otimes x_{1}$

(8)

$f_{1} \otimes f_{2} - 3\left(f_{2} \otimes f_{3}\right)$ .

Problem 36.16

Find the components of the tensors from items 1, 3, 5, 7, 8 of problem 36.15, if the vectors $x_{1}, x_{2}, x_{3}$ and the covectors $\mathrm{f}_{1}, \mathrm{f}_{2}, \mathrm{f}_{3}$ are given by means of columns and rows respectively: $\mathbf{c}_{20}, \mathbf{c}_{12}, \mathbf{c}_{25}, \mathbf{c}_{8}^{T}$ , $\mathbf{c}_{10}^{T}, \mathbf{c}_{22}^{T}$ . $\mathbf{c}_{20} = \left[\begin{smallmatrix} 0 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{25} = \left[\begin{smallmatrix} 2 \\ 4 \end{smallmatrix}\right], \mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{10} = \left[\begin{smallmatrix} -4 \\ 6 \end{smallmatrix}\right], \mathbf{c}_{22} = \left[\begin{smallmatrix} 14 \\ -9 \end{smallmatrix}\right]$ .

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Problem 36.17

(1)(2)(3)

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(1)

Let $a = x\otimes y$ , and let the vectors $x$ and $y$ have, in the basis $\mathbf{e}$ , components $1, 0, 0$ and $0, 1, 0$ respectively. Find the components of the tensor $a$ in the basis $e$ and in the basis $\mathbf{e}^{\prime } = \mathbf{e} S$ , where $S = \mathbf{A}_{207}$ . $\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right]$ .

(2)

Let $a = \mathrm{f} \otimes \mathrm{g}$ , and let the covectors $\mathrm{f}$ and $\mathrm{g}$ have, in the basis $e$ , coordinate rows $(1, 0, 0)$ and $(0, 1, 0)$ respectively. Find the components of the tensor $a$ in the basis $\mathbf{e}$ and in the basis $\mathbf{e}^{\prime } = \mathbf{e} S$ , where $S = \mathbf{A}_{207}$ . $\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right]$ .

(3)

Let $a = x\otimes \mathrm{f}$ , and let the vector $x$ and covector $\mathrm{f}$ have, in the basis, components $1, 0, 0$ and $0, 1, 0$ respectively. Find the components of the tensor $a$ in the basis $e$ and in the basis $\mathbf{e}^{\prime } = \mathbf{e} S$ , where $S = \mathbf{A}_{207}$ . $\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right]$ .

Compare the results of problems 1), 2), 3).

Problem 36.18

(1)(2)

Decompose the tensor into a product of univalent tensors, if it has:

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(1)

type $(2, 0)$ and matrix $\mathbf{A}_{5}$ . $\mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right]$ .

(2)

type $(2, 1)$ and matrix $\mathbf{A}_{673}$ . $\mathbf{A}_{673} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & -1 \\ 2 & -2 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & -1 \\ 2 & -2 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

Problem 36.19

(1)(2)

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(1)

Let $a$ be a tensor of type $(1, 1)$ whose component matrix has rank $r$ . Prove that there exist $r$ linearly independent vectors $a_{1}, \ldots , a_{r}$ and $r$ linearly independent covectors $\mathrm{f}_{1}, \ldots , \mathrm{f}_{r}$ such that $a = \sum_{\alpha = 1}^{r} a_{\alpha } \otimes \mathrm{f}^{\alpha }$ .

(2)

State and prove an analogous statement for tensors of type $(2, 0)$ .

Problem 36.20

(1)(2)(3)

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(1)

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

(2)

Represent the bilinear function $3\xi^{1} \eta^{1} + 2\xi^{1} \eta^{2} + 3\xi^{2} \eta^{1} + 2\xi^{2} \eta^{2}$ as a product of linear ones. Is such a representation unique?

(3)

A bilinear function $\mathrm{f}$ is given, in some basis of a vector space, by the matrix $\mathbf{A}_{454}$ . Represent it as a sum of two products of pairs of linear functions: $\mathrm{f}(x, y) = \mathrm{f}_{1}(x) \mathrm{g}_{1}(y) + \mathrm{f}_{2}(x) \mathrm{g}_{2}(y)$ . Is this representation unique? $\mathbf{A}_{454} = \left[\begin{smallmatrix} 2 & 4 & 6 & 1 \\ 1 & 2 & 3 & 1 \\ -3 & -6 & -9 & 1 \\ 1 & 2 & 3 & 1 \end{smallmatrix}\right]$ .

Subsection 36.2Collapse (36.21-36.29)[9/78%]

Problem 36.21

(1)(2)(3)(4)

Starting from the transformation laws of the tensors $a_{jk}^{i}, a_{j}^{i}$ , $b_{j}^{i}, a_{i}, b^{ij}, a_{klm}^{ij}, \xi^{k}$ , verify the transformation law of the components of the contractions:

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(1)

$a_{ji}^{i}$

(2)

$a_{j}^{i} b_{k}^{j}$

(3)

$a_{i} b^{ij}$

(4)

$a_{klm}^{ij} \xi^{k}$ .

Problem 36.22

(1)(2)(3)

Starting from the geometric meaning of the tensors $a_{i}, \xi^{i}$ , $a_{j}^{i}, b_{ij}$ , explain the geometric meaning of the contractions:

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(1)

$a_{i} \xi^{i}$

(2)

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

(3)

$b_{ij} \xi^{i} \xi^{j}$ .

Problem 36.23

(1)(2)(3)

Can one contract:

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(1)

a vector and a covector?

(2)

a vector and a vector?

(3)

a pair of covectors?

Problem 36.24

Write the product of linear transformations in tensor notation.

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Problem 36.25

(1)(2)(3)

Tensors $a_{j}^{i}, \xi^{i}, \varkappa_{i}$ are given by the matrices: $\mathbf{A}_{232}, \mathbf{c}_{104}$ , $\mathbf{c}_{104}^{T}$ . Compute the contractions: $\mathbf{A}_{232} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 1 & 1 & 1 \end{smallmatrix}\right], \mathbf{c}_{104} = \left[\begin{smallmatrix} 1 \\ -2 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{104}^{T} = \left[\begin{smallmatrix} 1 & -2 & 3 \end{smallmatrix}\right]$ .

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(1)

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

(2)

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

(3)

$a_{j}^{i} \xi^{j} \varkappa_{i}$

Problem 36.26

How many distinct tensors can be formed by contraction from a given tensor of type $(2, 2)$ ?

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Problem 36.27

(1)(2)

A tensor $a_{k}^{ij}$ is given by the matrix:

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(1)

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

(2)

$\mathbf{A}_{655}$ . $\mathbf{A}_{655} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 0 & -2 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 3 \\ 3 & 2 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

Find the matrices of the contractions:

(1) $a_{i}^{i j}$ ,

(2) $a_{j}^{i j}$ .

Problem 36.28

(1)(2)(3)

A tensor $a_{kl}^{ij}$ is given by the matrix:

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(1)

$\mathbf{A}_{649}$ . $\mathbf{A}_{649} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 5 \\ 2 & 6 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 3 & 7 \\ 4 & 8 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{693}$ . $\mathbf{A}_{693} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 0 & 0 \\ 0 & 0 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} -1 & -2 \\ -3 & -4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 0 \\ 0 & 0 \end{smallmatrix}\right]\end{smallmatrix}\right]$ .

(3)

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

Compute the contractions:

(1) $a_{i l}^{i j}$ ;

(2) $a_{k j}^{i j}$ ;

(3) $a_{k i}^{i j}$ ;

(4) $a_{i l}^{i j}$ ;

(5) $a_{i j}^{i j}$ ;

(6) $a_{j i}^{i j}$ .

Problem 36.29

(1)(2)

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(1)

To each basis of the space $\mathcal{L}_{n}$ is assigned an ordered set of numbers $a_{klm}^{ij}$ (all indices range over the values from 1 to $n$ ). It is known that for an arbitrary vector $\xi^{k}$ the numbers $a_{klm}^{ij} \xi^{k}$ are the components of a tensor of type $(2, 2)$ . Prove that $a_{klm}^{ij}$ is a tensor of type $(2, 3)$ .

(2)

To each basis of the space $\mathcal{L}_{n}$ is assigned an ordered set of numbers $a_{klm}^{ij}$ (all indices range over the values from 1 to $n)$ . It is known that for an arbitrary tensor $u_{ij}^{k}$ of type $(1, 2)$ the numbers $a_{klm}^{ij} u_{ij}^{k}$ are the components of a tensor of type $(0, 2)$ . Prove that $a_{klm}^{ij}$ is a tensor of type $(2, 3)$ .

Subsection 36.3Transposition, symmetrization, alternation. Symmetric and antisymmetric tensors (36.30-36.57).[28/71%]

Problem 36.30

(1)(2)

Can one transpose the tensor:

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(1)

of type $(1, 1)$

(2)

of type $(2, 0)$ ?

Problem 36.31

One tensor of type $(0, 2)$ is obtained from another by transposition. How are the corresponding bilinear functions related?

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Problem 36.32

(1)(2)(3)(4)

Tensors

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(1)

$a_{ij}$

(2)

$a^{ij}$

(3)

$a_{k}^{ij}$

(4)

$a_{jk}^{i}$

are given, respectively, by the matrices $\mathbf{A}_{16}, \mathbf{A}_{16}, \mathbf{A}_{670}, \mathbf{A}_{670}$ . Find the matrices of the transposed tensors. $\mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{670} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 5 & 6 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 3 & 4 \\ 7 & 8 \end{smallmatrix}\right] \end{smallmatrix}\right], \mathbf{A}_{670} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 5 & 6 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 3 & 4 \\ 7 & 8 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

Problem 36.33

(1)(2)(3)(4)

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(1)

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

(2)

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 $(3, 0)?$ . $\mathbf{A}_{670} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 5 & 6 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 3 & 4 \\ 7 & 8 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(3)

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 $b_{ijk} = a_{jki}, c_{ijk} = a_{ikj}$ . $\mathbf{A}_{727} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 3 & 2 \\ 9 & 7 & 8 \\ 4 & 2 & 3 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 7 & 9 & 8 \\ 3 & 1 & 2 \\ 8 & 6 & 7 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 4 & 6 & 5 \\ 6 & 4 & 5 \\ 1 & 9 & 5 \end{smallmatrix}\right]\end{smallmatrix}\right]$ .

(4)

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 $b_{ijkl} = a_{kjli}; c_{ijkl} = a_{lkji}$ . $\mathbf{A}_{717} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 5 & 6 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 3 & 4 \\ 7 & 8 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} 9 & 10 \\ 13 & 14 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 11 & 12 \\ 15 & 16 \end{smallmatrix}\right]\end{smallmatrix}\right]$ .

Problem 36.34

Let $a, b$ be tensors of type $(1, 1)$ . Express the tensor $c = b\otimes a$ in terms of $d = a\otimes b$ .

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Problem 36.35

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)

Without using the shortened notation, write out all the components of the tensors given in the space $\mathcal{L}_{2}$ :

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(1)

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

(2)

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

(3)

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

(4)

$x^{i} a_{jk}$

(5)

$x^{i} a_{ik}$

(6)

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

(7)

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

(8)

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

(9)

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

(10)

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

(11)

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

(12)

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

(13)

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

(14)

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

(15)

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

(16)

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

Problem 36.36

(1)(2)(3)(4)

A tensor $a^{ij}$ is given by the matrix:

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(1)

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

(2)

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

(3)

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

(4)

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

Find the components of the tensors:

(1) $a^{(i j)}$ ;

(2) $a^{\left[i j\right]}$ .

Problem 36.37

(1)(2)(3)

A tensor $a_{ijk}$ is given by the matrix:

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(1)

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

(2)

(3)

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

Find the components of the tensors:

(1) $a_{(i j) k}$ ;

(2) $a_{i(j k)}$ ;

(3) $a_{(i\left|j\right| k)}$ ;

(4) $a_{(i j k)}$

Problem 36.38

(1)(2)

A tensor $a_{kl}^{ij}$ is given by the matrix:

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(1)

(2)

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

Find the components of the tensors:

(1) $a_{k l}^{(i j)}$ ;

(2) $a_{(k l)}^{i j}$ ;

(3) $a_{(k l)}^{(i j)}$ .

Problem 36.39

(1)(2)(3)

A tensor $a_{ijk}$ is given by the matrix:

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(1)

(2)

(3)

Find the components of the tensors:

(1) $a_{\left[i j\right] k}$ ;

(2) $a_{i \left[j k\right]}$ ;

(3) $a_{\left[i \mid j \mid k\right]}$ .

Problem 36.40

(1)(2)

A tensor $a_{kl}^{ij}$ is given by the matrix:

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(1)

(2)

Find the components of the tensors:

(1) $a_{k l}^{\left[i j\right]}$ ;

(2) $a_{\left[k l\right]}^{i j}$ ;

(3) $a_{\left[k l\right]}^{\left[i j\right]}$ .

Problem 36.41

(1)(2)

A tensor $a_{ijk}$ is given by the matrix:

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(1)

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

(2)

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

Find the components of the tensors:

(1) $a_{\left[i j k\right]}$ ;

(2) $a_{(i j k)}$ .

Problem 36.42

(1)(2)(3)(4)(5)

A tensor of type $(0, 3)$ is given by the matrix:

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(1)

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

(2)

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

(3)

(4)

(5)

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

Determine whether the tensor is symmetric (antisymmetric), and if so, with respect to which indices.

Problem 36.43

(1)(2)

A tensor $a_{j}^{i}$ is given by the matrix $A$ :

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(1)

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

(2)

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

Compute the invariants:

(1) $a_{i}^{i}$ ;

(2) $a^{i}_{[i} a^{k}_{k]}$ ;

(3) $a^{i}_{[i} a^{j}_{j} a^{k}_{k]}$ .

Compare the invariants found with the coefficients of the characteristic polynomial of the matrix $A$ .

Problem 36.44

(1)(2)(3)(4)(5)

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(1)

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

(2)

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

(3)

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.

(4)

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.

(5)

Prove statement 4) for lower indices.

Problem 36.45

Let $a_{ij}$ and $b^{ij}$ be the components of a symmetric and an antisymmetric tensor, respectively. Compute the contraction $a_{ij} b^{ij}$ .

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Problem 36.46

(1)(2)

For the tensor $\delta_{j_{1} \ldots j_{k}}^{i_{1} \ldots i_{k}}$ , defined in problem 35. 20, and arbitrary tensors $a^{j_{1} \ldots j_{k}}$ and $b_{i_{1} \ldots i_{k}}$ , prove that:

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(1)

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

(2)

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

Problem 36.47

Let $a_{ijk} \xi^{i} \xi^{j} \xi^{k} = 0$ for any vector $\xi^{i}$ . Prove that $a_{\left[ijk\right]} = 0$ .

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Problem 36.48

Prove that $a_{j}^{(i} a_{l}^{k)} = a_{(j}^{i} a_{l)}^{k}$ , $a_{j}^{[i} a_{l}^{k]} = a_{[j}^{i} a_{l]}^{k}$ .

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Problem 36.49

(1)(2)(3)(4)(5)

Compute:

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(1)

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

(2)

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

(3)

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

(4)

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

(5)

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

Problem 36.50

(1)(2)

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(1)

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.

(2)

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.

Problem 36.51

(1)(2)

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(1)

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

a_{(ijk)} = \frac{1}{3}\left(a_{ijk} + a_{kij} + a_{jki}\right)

(2)

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

a_{[ijk]} = \frac{1}{3}\left(a_{ijk} + a_{kij} + a_{jki}\right)

Problem 36.52

(1)(2)(3)

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(1)

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.

(2)

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.

(3)

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.

Problem 36.53

(1)(2)(3)

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(1)

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.

(2)

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.

(3)

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.

Problem 36.54

Prove that any tensor of type $(0, 2)$ or $(2, 0)$ can be decomposed into a sum of a symmetric and an antisymmetric tensor.

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Problem 36.55

(1)(2)(3)

Decompose into a sum of a symmetric and an antisymmetric tensor the tensor of type $(0, 2)$ given by the matrix:

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(1)

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

(2)

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

(3)

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

Problem 36.56

(1)(2)

From the Kronecker symbol, using tensor operations, obtain the tensors:

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(1)

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

(2)

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

Problem 36.57

(1)(2)(3)

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(1)

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 $a = \sum_{\alpha = 1}^{r} \mathrm{f}_{\alpha } \otimes \mathrm{f}_{\alpha }$ .

(2)

State and prove the converse statement.

(3)

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? $\mathbf{A}_{56} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 3 \end{smallmatrix}\right]$ .