\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}} \renewcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\t}[1]{\textsf{#1}} \newcommand{\m}[1]{\mathbb{#1}} \def\RR{\bf R} \def\bold#1{\bf #1} \def\mbf#1{\mathbf #1} \def\uv#1{\hat{\usf {#1}}} \def\dl#1{\underline{\underline{#1}}} \newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}} \def\bs#1{\usf #1}\]

Transpose, Symmetric and Skew-Symmetric Tensors; Symmetric and Skew-symmetric parts of a tensor

Definition: The transpose of a tensor \(\bs{T}\) is the tensor \(\bs{T}^{\rm T}\) that obeys the following rule

\[\begin{align} \bs{u}\cdot\left(\bs{T}\bs{v}\right)&= \bs{v}\cdot\left(\bs{T}^{\rm T}\bs{u}\right) \end{align}\]

where \(\bs{v}\) and \(\bs{u}\) are arbitrary vectors in \(\mathbb{E}^N\).

\[\begin{align} [\bs{T}^{\rm T}]_{ij} = [\bs{T}]_{ji} \end{align}\]

Symmetric Tensor

Definition: A tensor is symmetric if it is equal to its transpose.

If \(T\) is symmetric then

\[\begin{align} \bs{T}&=\bs{T}^{\text{T}}, \end{align}\]

Recall the definition of transpose:

\[\begin{align} \bs{x}\cdot\bs{T}^{\rm T}\bs{y} &= \bs{y}\cdot\bs{T}\bs{x} \\ \bs{x}\cdot\bs{T}\bs{y} &= \bs{y}\cdot\bs{T}\bs{x} \end{align}\]

If \(\bs{x} = \hat{\bs{E}}_{i}, \bs{y} = \hat{\bs{E}}_{j}\), then

\[\begin{align} \hat{\bs{E}}_{i}\cdot(\bs{T}\hat{\bs{E}}_{j}) = \hat{\bs{E}}_{j}\cdot(\bs{T}\hat{\bs{E}}_{i}) \end{align}\]

Recall

\[T_{pq} = [\bs{T}]_{pq} = \hat{\bs{E}}_{p}\cdot(\bs{T}\hat{\bs{E}}_{q})\]

thus

\[\begin{align} T_{ij} = T_{ji} \end{align}\]

Symmetric tensor can be written as

\[\begin{align} \textsf{sym}(\bs{T}) = \frac{1}{2}(\bs{T}+\bs{T}^{\rm T}). \end{align}\]

Skew-Symmetric Tensor

Definition: A tensor is skew-symmetric if it is equal to the negative of its transpose.

If \(T\) is skew-symmetric then

\[\begin{align} \bs{T}&=-\bs{T}^{\text{T}}, \end{align}\]

In component form,

\[\begin{align} \bs{x}\cdot\bs{T}\bs{y} &= \bs{y}\cdot\bs{T}^{\rm T}\bs{x} \\ \bs{x}\cdot\bs{T}\bs{y} &= -(\bs{y}\cdot\bs{T}\bs{x}) \\ \hat{\bs{E}}_{i}\cdot(\bs{T}\hat{\bs{E}}_{j}) &= -(\hat{\bs{E}}_{j}\cdot(\bs{T}\hat{\bs{E}}_{i})) \\ T_{ij} &= -T_{ji} \end{align}\]

Skew symmetric tensor can be written as

\[\begin{align} \textsf{skew}(\bs{T}) = \frac{1}{2}(\bs{T}-\bs{T}^{\rm T}). \end{align}\]

Properties of Transposes

(i) \((\bs{A}^{\rm T})^{\rm T} = \bs{A}\)

(ii) \((\bs{A}\bs{B})^{\rm T} = \bs{B}^{\rm T}\bs{A}^{\rm T}\)

(iii) \((\bs{A}+\bs{B})^{\rm T} = \bs{A}^{\rm T} + \bs{B}^{\rm T}\)

(iv) \((\alpha\bs{A})^{\rm T} = \alpha \bs{A}^{\rm T}\)

In component form,

(i) \([[\bs{A}^{\rm T}]_{ij}]^{\rm T} = [[\bs{A}]_{ji}]^{\rm T} = [\bs{A}^{\rm T}]_{ji} = [\bs{A}]_{ij}\)

(ii)

\[\begin{align} [\bs{B}^{\rm T}\bs{A}^{\rm T}]_{ij} &= [\bs{B}]^{\rm T}_{ik}[\bs{A}]^{\rm T}_{kj} \\ &= [\bs{B}]_{ki}[\bs{A}]_{jk} \\ &= [\bs{A}\bs{B}]_{ji} \\ &= [\bs{A}\bs{B}]^{\rm T}_{ij} \end{align}\]

(iii)

\[\begin{align} [\bs{A}^{\rm T}+\bs{B}^{\rm T}]_{ij} &= [\bs{A}]^{\rm T}_{ij} + [\bs{B}]^{\rm T}_{ij} \\ &= [\bs{A}]_{ji} + [\bs{B}]_{ji} \\ &= [\bs{A} + \bs{B}]_{ji} \\ &= [\bs{A} +\bs{B}]^{\rm T}_{ij} \end{align}\]

(iv)

\[\begin{align} \alpha[\bs{A}^{\rm T}]_{ij} &= \alpha[\bs{A}]_{ji} \\ &= [\alpha\bs{A}]^{\rm T}_{ij} \end{align}\]

Note:

If \(a_{i}\hat{\bs{E}_{i}} = b_{j}\hat{\bs{E}_{j}}\), then is it true that \(a_{i} = b_{i}?\)

\[\begin{align} & a_{i}\hat{\bs{E}_{i}}\cdot\hat{\bs{E}_{p}} = b_{j}\hat{\bs{E}_{j}}\cdot\hat{\bs{E}_{p}} \\ & a_{i}\delta_{ip} = b_{j}\delta_{jp} \\ & \boxed{a_{p} = b_{p}} \end{align}\]

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