Tensors

A tensor of type $(1,0)$ has components in the basis $e_{1}, \ldots , e_{n}$ ($i_{0}$ is a fixed number, $1 \leqslant i_{0} \leqslant n$)

Find the components of this tensor in the new basis $e_{1}^{\prime }, \ldots , e_{n}^{\prime }$, given the transition matrix $C = \left(c_{j}^{i}\right)$ from the old basis to the new one.

A tensor of type $(0,1)$ has components in the basis $e_{1}, \ldots , e_{n}$ ($i_{0}$ is a fixed number, $1 \leqslant i_{0} \leqslant n$)

Find the components of this tensor in the new basis $e_{1}^{\prime }, \ldots , e_{n}^{\prime }$, given the transition matrix $C = \left(c_{j}^{i}\right)$ from the old basis to the new one.

Verify that a vector (respectively a covector (linear function), a linear operator, a bilinear form) on $V$ defines a tensor of type $(1,0)$ (respectively $(0,1),(1,1),(0,2)$).

Verify that the maps defined above between the spaces of tensors $\mathbf{T}_{q}^{p}$ and multilinear functions $\mathbf{P}_{q}^{p}$ are well-defined, linear, and bijective. How is the operation $\otimes$ written in terms of $\mathbf{P}_{q}^{p}$?

Verify that the tensor product and contraction are well-defined tensor operations.

Show that assigning the Kronecker symbol $\delta_{j}^{i}$ to every basis defines a tensor of type $(1,1)$. Which linear operator and which multilinear function correspond to it?

Show that $\varepsilon_{i_{1} \ldots i_{n}}$ is not a tensor.

Let $v \in V, \varphi \in V^{*}$. Find the image of the operator corresponding to the tensor $v \otimes \varphi$ in the space $V$.

Fix a basis $e_{1}, e_{2}, e_{3}$ in $V$ and define the tensor $A = e^{1} \otimes e_{2}-e^{2} \otimes e_{3}+e^{3} \otimes e_{1} \in \mathbf{T}_{1}^{1}$ and the trilinear map $F \in \mathbf{P}_{3}^{0}$, $F\left(v_{1}, v_{2}, v_{3}\right) = \det \left\| v_{j}^{i}\right\|$, where $v_{j}^{i}$ is the $i$-th coordinate of the vector $v_{j}$ in the given basis. Compute $\left(A \otimes T_{F}\right)_{2133}^{1}$ and $\left(T_{F} \otimes A\right)_{2133}^{1}$.

Let, in the basis $e_{1}, e_{2}, e_{3}$, the components of the tensor $T \in \mathbf{T}_{3}^{2}(V)$ be defined as $T_{k s m}^{i j} = 2 i$. Find $\left(T^{\prime }\right)_{223}^{12}$, if $e_{i}^{\prime } = \sum_{j \leqslant i}(i-j+1) e_{j}$.

Show that a tensor of type $(1,1)$, invariant under all orthogonal changes of coordinates in $\mathbb {R}^{n}$, is proportional to the tensor $\delta_{j}^{i}$.

Show that a tensor of the third valence, invariant under arbitrary changes of coordinates, is equal to zero.

Find the general form of a tensor of the fourth valence, invariant under an arbitrary change of coordinates. Consider the various cases $(p, q), p+q = 4$.

Express the trace of the operator A as the result of tensor operations.

Express the determinant of the matrix of an operator as the result of tensor operations.

Let $A$ be a linear operator. Find expressions for the quantities $A_{i}^{i}, A_{j}^{i} A_{i}^{j}, A_{j}^{i} A_{k}^{j} A_{i}^{k}$ in terms of the roots and coefficients of the characteristic polynomial $\operatorname {det}\left(A-\lambda E\right)$.

Consider the multilinear function $F \in \mathbf{P}_{n}^{n}(V), \operatorname {dim} V = n$, given by the formula

For a given basis $e_{1}, \ldots , e_{n}$, find the components of $T_{F}$.

Let $\varphi \in \mathbf{P}_{q}^{p}$. Show that the value of $\varphi$ on a specific set of covector arguments $\xi^{j}$ and vector arguments $v_{i}$, i.e. $\varphi \left(\xi^{1}, \ldots , \xi^{p}, v_{1}, \ldots , v_{q}\right)$, coincides with the $(p+q)$-fold contraction of the tensor $T_{\varphi } \otimes \xi^{1} \otimes \ldots \otimes \xi^{p} \otimes v_{1} \otimes \ldots \otimes v_{q}$ of type $(p+q, p+q)$.

Let $\mathbb {R}_{n}[x]$ be the space of all polynomials of degree at most $n$ in the variable $x$ with real coefficients. Verify that the map $l: \mathbb {R}_{n}[x] \rightarrow \mathbb {R}$, given by each of the following methods:

Verify that $V \otimes W$ is a linear space.

Let $\mathbb {k}[x]$ be the space of all polynomials in the variable $x$ with coefficients in the field $\mathbb {k}$, and let $\mathbb {k}[x, y]$ be the space of polynomials in $x$ and $y$. Prove that the correspondence $f \otimes g \mapsto f g$ defines a linear isomorphism $\mathbb {k}[x] \otimes \mathbb {k}[y] \rightarrow \mathbb {k}[x, y]$.

Let $\mathbb {k}_{n}[x]$ be the space of all polynomials of degree at most $n$ in the variable $x$ with coefficients in the field $\mathbb {k}$, and let $\mathbb {k}_{n, m}[x, y]$ be the space of polynomials in $x$ and $y$ of degree at most $n$ in $x$ and at most $m$ in $y$. Prove that the correspondence $f \otimes g \mapsto f g$ defines a linear isomorphism $\mathbb {k}_{n}[x] \otimes \mathbb {k}_{m}[y] \rightarrow \mathbb {k}_{n, m}[x, y]$.

Prove the associativity and commutativity (up to natural isomorphisms) of the tensor product operation on spaces.

Establish the natural isomorphism

Consider $F \in \mathbf{P}_{q+q^{\prime }}^{p+p^{\prime }}$. Prove that the map

defines a linear map $\widetilde{F}: \mathbf{P}_{p}^{q} \rightarrow \mathbf{P}_{q^{\prime }}^{p^{\prime }}$.

Let $\operatorname {dim} V = n, \operatorname {dim} W = m$, and let $e_{1}, \ldots , e_{n}$ and $f_{1}, \ldots , f_{m}$ be some bases of the spaces $V$ and $W$. Prove that $\operatorname {dim}(V \otimes W) = n \cdot m$, and that $e_{i} \otimes f_{j}, i = 1, \ldots , n, j = 1, \ldots , m$ can be chosen as a basis.

Let there be given decompositions of spaces into direct sums: $V = \bigoplus_{i} V_{i}, W = \bigoplus_{j} W_{j}$. Prove that $V \otimes W \cong \bigoplus_{i, j} V_{i} \otimes W_{j}$ (after the natural identification).

In a special case, the isomorphism between the tensor product and the space of tensors (see problem 1706) becomes the canonical isomorphism $\tau : V^{*} \otimes V \rightarrow \mathscr {L}(V)$, where $\mathscr {L}(V)$ is the space of linear operators. Compute (first justifying independence of the dimension):

In the notation of the previous problem, find, for $\operatorname {dim} V = 4$

where $\circ$ denotes composition of maps.

Let $V = \mathbb {R}_{n}[x]$ be the space of all polynomials of degree at most $n$ in the variable $x$ with real coefficients, and let the function $l: V \rightarrow \mathbb {R}$, given by the formula

define a covector. Also consider the polynomials $r = 1-x+x^{2}$ and $q = 1+x+x^{2}+\ldots +x^{n}$. Find $\tau (l \otimes r)(q)$.

Show that $f \otimes g$, as defined in the theoretical part, is a well-defined linear map.

Let us denote $V^{\otimes p} = \underbrace{V \otimes \ldots \otimes V}_{p}$ and $f^{\otimes p} = \underbrace{f \otimes \ldots \otimes f}_{p}$, where $f$ is a linear map. Prove that $\operatorname {rk}\left(f^{\otimes p}\right) = (\operatorname {rk}\left(f\right))^{p}$. In particular, the tensor power of an invertible operator is invertible.

For operators in the space $V$, prove the formulas:

Prove that $\operatorname {tr}\left(f^{\otimes p}\right) = (\operatorname {tr}(f))^{p}$, where $f$ is a linear operator.

Let $e_{1}, \ldots , e_{p}$ be (not necessarily distinct) eigenvectors of the operator $f: V \rightarrow V$, corresponding to the eigenvalues $\lambda_{1}, \ldots , \lambda_{p}$ respectively. Prove that $e_{1} \otimes \ldots \otimes e_{p}$ is an eigenvector of the operator $f^{\otimes p}$, corresponding to the eigenvalue $\lambda_{1} \cdot \ldots \cdot \lambda_{p}$.

Let the linear operator $f: V \rightarrow V$ be reducible to diagonal form. Prove that in this case $f^{\otimes p}$ is also diagonalizable. Is the converse statement true?

Let $\lambda_{k}(k = 1, \ldots , n)$ be all the eigenvalues of the operator $f: V \rightarrow V$, taken with multiplicity, $\operatorname {dim} V = n$. Prove that the set of all eigenvalues of the operator $f^{\otimes p}$, taken with multiplicity, consists of all possible products

Prove that $\operatorname {det}\left(f^{\otimes p}\right) = (\det f)^{p n^{p-1}}$, where $f$ is an operator in $V$, $\operatorname {dim} V = n$.

Let the linear operators $f: V \rightarrow V$ and $g: W \rightarrow W$ have corresponding decompositions into invariant subspaces $V = \bigoplus_{i} V_{i}, W = \bigoplus_{j} W_{j}$. Prove that $V \otimes W = \bigoplus_{i, j} V_{i} \otimes W_{j}$ is a decomposition into invariant subspaces for $f \otimes g$.

Let $\frac{d}{d x}$ and $\frac{d}{d y}$ be the differentiation operators in the spaces of polynomials $\mathbb {R}_{n}[x]$ and $\mathbb {R}_{m}[y]$ respectively. Determine the analytic meaning of the operator $\frac{d}{d x} \otimes \frac{d}{d y}$ and find its Jordan form.

Find the Jordan form of the tensor product of two operators having the following Jordan forms:

Verify that the operators Sym and Alt in the space $\mathbf{T}_{0}^{p}$ have the following properties (where $\circ$ denotes composition of maps):

Show that $S(V)$ with the operation $\bullet$ and $\Lambda (V)$ with the operation $\wedge$ are associative algebras. Moreover:

Let $e_{1}, \ldots , e_{n}$ be a basis of the space $V$. Prove that the tensors

form a basis of the space $S^{p}(V)$. Compute its dimension.

Let $e_{1}, \ldots , e_{n}$ be a basis of the space $V$. Prove that the tensors

form a basis in $\Lambda^{p}(V)$. Compute its dimension and the dimension of the space $\Lambda (V)$. Carry out analogous reasoning for $\Omega_{q}(V)$.

Prove that the linear span of the set

equals $\mathbf{S}^{k}(V)$ (the field $\mathbb {k}$ has characteristic 0).

Prove that the natural embeddings

induce isomorphisms

Let $\operatorname {dim} V > 2$. Show that the spaces $\Lambda^{2}\left(\Lambda^{2}(V)\right)$ and $\Lambda^{4}(V)$ are not isomorphic.

Show that the exterior product $a_{1} \wedge a_{2} \wedge \ldots \wedge a_{p}$ of the vectors $a_{1}, a_{2}, \ldots , a_{p}$ equals zero if and only if these vectors are linearly dependent.

Let $T$ be a skew-symmetric covariant tensor of valence $n$ in the space $V$, with $n = \operatorname {dim} V$. Show that all its components coincide, up to sign, with $T_{12 \ldots n}$ (one says that the tensor $T$ has a single essential component). Prove that this component transforms according to the rule

where $\det C$ is the determinant of the transition matrix to the new basis. Similarly, in the contravariant case, prove the formula $\left(T^{\prime }\right)^{12 \ldots n} = (\det C)^{-1} \cdot T^{12 \ldots n}$.

Let $a_{1}, \ldots , a_{p}$ and $b_{1}, \ldots , b_{p}$ be two sets of linearly independent vectors. Prove that $a_{1} \wedge \ldots \wedge a_{p} = \lambda \cdot b_{1} \wedge \ldots \wedge b_{p}$ if and only if the linear spans of the two sets of vectors coincide. Moreover, $\lambda$ equals the determinant of the transition matrix from the second system of vectors to the first.

Show that the linear span of a system of linearly independent vectors $a_{1}, \ldots , a_{k}$ coincides with the set of those vectors $a$ for which $T \wedge a = 0$, where $T = a_{1} \wedge \ldots \wedge a_{k}$.

Let $T \in \Lambda^{p}(V)$; define

Prove that $T$ can be represented in the form $\widetilde{T} \wedge S$, where $S = a_{1} \wedge \ldots \wedge a_{k} \neq 0$, if and only if $a_{i} \in \operatorname {Ann}(T)$ for $i = 1, \ldots , k$.

Prove that the correspondence $T \mapsto \operatorname {Ann}(T)$ is a bijection from the set of nonzero decomposable $p$-vectors in $V$ (considered up to multiplication by a scalar) onto the set of $p$-dimensional subspaces of the space $V$.

Show that for $T \in \Lambda^{p}(V)$ the dimension of the space $\operatorname {Ann}(T)$ does not exceed $p$, with equality holding precisely when $T$ is the exterior product of linearly independent vectors.

Let a skew-symmetric tensor $T \in \Lambda^{n-1}(V)$ be given, where $\operatorname {dim} V = n$. Prove that there exist vectors $a_{1}, \ldots , a_{n-1}$ such that $T = a_{1} \wedge \ldots \wedge a_{n-1}$ (i.e. the $(n-1)$-vector $T$ is decomposable). Similarly, for $S \in \Omega_{n-1}(V)$ there exist covectors $\xi^{1}, \ldots , \xi^{n-1}$ such that $S = \xi^{1} \wedge \ldots \wedge \xi^{n-1}$.

Prove that a bivector $T$ can be represented in the form $a \wedge b$ for some vectors $a$ and $b$ if and only if $T \wedge T = 0$.

Let $\varphi$ be a nondegenerate bilinear function on $V$. Prove that on the space $\Lambda^{2}(V)$ one can define a nondegenerate bilinear form $\Phi$ such that for any elements $v_{i} \in V$, $i = 1, \ldots , 4$, we have

Let $f: V \rightarrow V$ be a linear operator. Show that the operator $f^{\otimes p}$ (the tensor power) commutes with the operator Alt, thereby defining a linear map $f^{\wedge p}: \Lambda^{p}(V) \rightarrow \Lambda^{p}(V)$.

Let $\operatorname {dim} V = n, f: V \rightarrow V$ be a linear operator. Prove that the operator $f^{\wedge (n-1)}$ is either equal to 0, or nondegenerate, or has rank 1.

Let $f: V \rightarrow V$ be a linear operator, $\operatorname {dim} V = n$. Since $\operatorname {dim} \Lambda^{n}(V) = 1$, the operator $f^{\wedge n}$ is multiplication by a scalar. Prove that this scalar equals $\det f$.

Prove the formulas ($\circ$ is composition):

Let $e_{1}, \ldots , e_{p}$ be linearly independent eigenvectors of the operator $f: V \rightarrow V$, corresponding respectively to the eigenvalues $\lambda_{1}, \ldots , \lambda_{p}$ (not necessarily distinct). Prove that $e_{1} \wedge \ldots \wedge e_{p}$ is an eigenvector of the operator $f^{\wedge p}$, corresponding to the eigenvalue $\lambda_{1} \cdot \ldots \cdot \lambda_{p}$.

Let the linear operator $f: V \rightarrow V$ be reducible to diagonal form. Prove that in this case $f^{\wedge p}$ is also diagonalizable. Is the converse true?

Let $\lambda_{k}(k = 1, \ldots , n)$ be all the eigenvalues of the operator $f: V \rightarrow V$, taken with multiplicity, $\operatorname {dim} V = n$. Prove that the set of all eigenvalues of the operator $f^{\wedge p}$, taken with multiplicity, consists of all possible products

Prove that $\operatorname {det}\left(f^{\wedge p}\right) = (\det f)^{\binom {n-1}{p-1}}$, where $f: V \rightarrow V$ is a linear operator, $\operatorname {dim} V = n$.

Let the characteristic polynomial of the operator $f: V \rightarrow V$, $\operatorname {dim} V = n$, equal $\sum_{p = 0}^{n} a_{p} \lambda^{n-p}$. Prove the equality $\operatorname {tr}\left(f^{\wedge p}\right) = (-1)^{n-p} a_{p}$.

Find the traces of the operators $f^{\wedge p}$ for nontrivial $\Lambda^{p}(V)$, if $f$ is given by its matrix:

Prove that if $\operatorname {tr}\left(f^{\wedge p}\right) = 0$ for every $p = 1,2, \ldots$, then the operator $f: V \rightarrow V$ is nilpotent.

Find the Jordan form of $f^{\wedge 2}$ from the known Jordan form of $f$:

Let $W$ be a $k$-dimensional subspace of $V$. Describe the natural embedding of $\Lambda^{p}(W)$ into $\Lambda^{p}(V)$. Prove that the operator $f: V \rightarrow V$ has $W$ as an invariant subspace if and only if $\Lambda^{k}(W)$ is invariant under $f^{\wedge k}$.

For a covector $\varphi \in V^{*}$, define the interior product operator as the map

where, as above, $F_{T} \in \mathbf{P}_{0}^{p}$ is the multilinear function corresponding to $T \in \mathbf{T}_{0}^{p}$. Prove that the interior product operator takes symmetric tensors to symmetric ones, and skew-symmetric ones to skew-symmetric ones. Let $\varphi_{1}$ and $\varphi_{2}$ be two covectors. Show that the operators $i\left(\varphi_{1}\right)$ and $i\left(\varphi_{2}\right)$ commute in $S(V)$ and anticommute in $\Lambda (V)$.

Let $\varphi$ be a nondegenerate bilinear function. Prove that assigning to each basis the matrix inverse to the matrix of $\varphi$ in this basis defines a tensor of type (2,0).

Represent the operations of raising and lowering an index (standing in any position) in terms of product and contraction. Thereby verify that raising and lowering are well-defined tensor operations.

Show that if we restrict ourselves to orthogonal changes of coordinates in a Euclidean space, then the transformation law for upper indices coincides with the transformation law for lower indices. More precisely, if in some orthogonal coordinate system $T^{i_{1} \ldots i_{p}} = T_{i_{1} \ldots i_{p}}$ for all $i_{1}, \ldots , i_{p}$, then they are also equal in any other orthogonal coordinate system.

Let, in the basis $e_{1}, e_{2}, e_{3}$, the matrix of the scalar product equal $\left(\begin{smallmatrix} 2 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2\end{smallmatrix}\right)$. Perform the lowering and raising of indices for the tensors (coordinates given in the indicated basis):

In the space $\mathbb {R}_{n}[x]$ of all polynomials of degree at most $n$ in the variable $x$ with real coefficients, consider the scalar product

Let $e_{0} = 1, e_{1} = x, e_{2} = x^{2}, \ldots , e_{n} = x^{n}$ be a basis. Find the coordinates, in the dual basis, of the covector obtained by raising the index of $q = 1+2 x+3 x^{2}$, for $n = 2$.

Let, in the basis $(e_{i})$, the matrix of the Euclidean scalar product in the space $V$ equal $\left\| g_{i j}\right\|$, and let the equation $a_{1} x^{1}+\ldots +a_{n} x^{n} = 0$ define an $(n-1)$-dimensional subspace (hyperplane) $W$. Prove that the expressions $v^{i} = g^{i j} a_{j}$, regarded as the components of some vector in this basis, define a normal vector to $W$.