Multilinear Algebra and Tensors
\(\require{physics}\def\vs#1{\mathsf{#1}}\)
Definition 63 (Covector)
A covector (linear form, one-form, or linear functional) is a linear map from a vector space to its field of scalars.
Definition 64 (Dual vector space)
The set of all covectors which map a vector space \(\vs{V}\) to its field \(F\) form a vector space over \(F\), which is called the dual space of \(\vs{V}\), and denoted \(\vs{V^{*}}\).
Suppose that vectors in the real vector space \(\mathbb{R}^n\) are represented as column matrices,
\[\begin{split}\vec{x} =
\begin{pmatrix}
x_1 \\ x_2 \\ \vdots \\ x_n
\end{pmatrix}\end{split}\]
Then any linear functional can be written in these coordinates as the sum
\[f(\vec{x}) = a_1 \vec{x}_1 + \cdots + a_n \vec{x}_n\]
can also be represented as a row matrix,
\[f =
\begin{pmatrix}
a_1, & a_2, & \cdots, & a_n
\end{pmatrix}\]
Definition 65 (Contravariant tensor)
Suppose \(n\) quantities in a basis \(\vec{x}^{1}, \dots, \vec{x}^n\) are related to another \(n\) quantities, :math:`A^{prime}^1, dots,
A^{prime}^n` in a coordinate system \(x^{\prime}^1, \dots, x^{\prime}^n\) by a set of transformations
\[A^{\prime}^p = \pdv{x^{\prime p}}{x^{\prime q}} A^q\]
Then \(A^1, \dots, A^n\) are the components of a first rank contravariant tensor.
Definition 66 (Covariant tensor)
Suppose \(n\) quantities in a basis \(\vec{x}_{1}, \dots, \vec{x}_n\) are related to another \(n\) quantities, :math:`A^{prime}_1, dots,
A^{prime}_n` in a coordinate system \(x^{\prime}_1, \dots, x^{\prime}_n\) by a set of transformations
\[A^{\prime}_p = \pdv{x^{\prime p}}{x^{\prime q}} A_q\]
Then \(A_1, \dots, A_n\) are the components of a first rank covariant tensor.
An intuitive introduction to tensors is via dyads, which were historically the precursors of tensors.
Definition 67 (Dyadic product)
Consider two vectors, \(\vec{u}, \vec{v} \in \vs{V}\), then their dyadic product, denoted \(\dyad{uv}\), can be represented by the sum
\[\begin{split}\begin{aligned}
\label{eq:dyadicproduct}
\dyad{ab} = & a_1 b_1 \dyad{ii} + a_2 b_1 \dyad{ij} + a_1 b_3 \dyad{ik} \\
+ & a_2 b_1 \dyad{ji} + a_2 b_2 \dyad{jj} + a_2 b_3 \dyad{jk} \\
+ & a_3 b_1 \dyad{ki} + a_3 b_2 \dyad{kj} + a_3 b_3 \dyad{kk}
\end{aligned}\end{split}\]
with the standard basis Dyads being of the form
\[\begin{split}\begin{matrix}
\dyad{ii} =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} &
\dyad{ij} =\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} &
\dyad{ik} =\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \\
\dyad{ji} =\begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} &
\dyad{jj} =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} &
\dyad{jk} =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \\
\dyad{ki} =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} &
\dyad{kj} =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} &
\dyad{kk} =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}
\end{matrix}\end{split}\]
In terms of dyads, both the outer product, \(\vec{a} \vec{b}^{\rm T}\) and the tensor product, \(\vec{a} \otimes \vec{b}\) are the same quantity.
Definition 68 (Tensor)
A tensor of type \((n, m-n)\) is an assignment of a multi-dimensional array,
\[T_{i_{n+1} \cdots i_m}^{i_1 \cdots i_n} \qty[\vec{f}]\]
to each basis \(\vec{f} = (\vec{e}_1, \dots \vec{e}_n)\), such that, if a change of basis is applied,
\[\vec{f} \to R \vec{f} = (R^i_1 \vec{e}_1, \dots, R^i_n \vec{e}_n )\]
then the array obeys the transform law
\[T_{i_{n+1} \cdots i_m}^{i_1 \cdots i_n} \qty[ \vec{f} \cdot R] = (R^{-1})^{i_1}_{j_1} R^{j_{n+1}}_{i_{n+1}} \cdots R^{j_m}_{i_m} T_{j_{n+1}, \dots, j_m}^{j_1 \dots j_n} \qty[\vec{f}]\]
Example 8 (The Stress Tensor)
The Stress Tensor is a quantity which involves forces acting on a three-dimensional object.
There are three forces, one acting on a plane perpendicular to each plane in the object, and each force having three spatial components.
The components of the force on the plane for which \(x\) is the normal are \(f_{xx}, f_{xy}, f_{xz}\), so the overall force on the body is
\[\begin{split}\begin{pmatrix}
f_{xx} & f_{xy} & f_{xz} \\ f_{yx} & f_{yy} & f_{yz} \\ f_{zx} & f_{zy} & f_{zz}
\end{pmatrix}\end{split}\]
and converting to stress, by dividing through by the area of each plane,
\[\begin{split}\begin{pmatrix}
\sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz}
\end{pmatrix}\end{split}\]
The elements of the leading diagonal are three orthogonal normal, or compressive stresses.
The six off-diagonal elements are orthogonal sheer stresses, so an alternative notation of the tensor is
\[\begin{split}\begin{pmatrix}
\sigma_x & \tau_{xy} & \tau_{xz} \\
\tau_{yx} & \sigma_y & \tau_{yz} \\
\tau_{xz} & \tau_{yz} & \sigma_z
\end{pmatrix}\end{split}\]
Tensor Rotations
Rotation tensors are an important quantity in physics, allowing the
transformation between coordinate systems. Let \((x,y,z)\) be a
coordinate system, and a second coordinate system, \((x^{\prime},
y^{\prime}, z^{\prime})\) is rotated relative to it. The rotation can be
described by a tensor of the form
\[\begin{split}\label{eq:1}
\begin{pmatrix}
x^{\prime} \\ y^{\prime} \\ z^{\prime}
\end{pmatrix} =
\begin{pmatrix}
\cos(\theta_{xx^{\prime}}) & \cos(\theta_{yx^{\prime}}) & \cos(\theta_{zx^{\prime}})\\
\cos(\theta_{xy^{\prime}}) & \cos(\theta_{yy^{\prime}}) & \cos(\theta_{zy^{\prime}})\\
\cos(\theta_{xz^{\prime}}) & \cos(\theta_{yz^{\prime}}) & \cos(\theta_{zz^{\prime}})
\end{pmatrix}
\begin{pmatrix}
x \\ y \\ z
\end{pmatrix}\end{split}\]
Using the summation convention, and letting the rotation matrix from
equation ([eq:1]) be \(a_{ij}\), we can rewrite the rotation
operation as
\[x_j^{\prime} = a_{ij} x_j\]
where \(a_{ij} = \hat{e}_i^{\prime}
\cdot \hat{e}_j\). The rotation is orthonormal, so
\(A^{-1} = A^{\rm T}\), and so
\[x_i = a_{ji} x_j^{\prime}\]
Extensions of Rotations to rank-2 Tensors
A tensor can be produced from the outer product of two vectors,
Definition 69 (Vector outer product)
Let \(\vec{u}\) and \(\vec{v}\) be vectors, then
\[\vec{u} \otimes \vec{v} = \vec{u} \vec{v}^{\rm T}\]
is the outer product of the two vectors, and is a rank-2 tensor.
Now, suppose the vectors \(\vec{u}\), and \(\vec{v}\) are rotated to become
\[\begin{split}\begin{aligned}
u_i^{\prime} &= a_{ik} u_k \\
v_j^{\prime} &= a_{jl} v_l
\end{aligned}\end{split}\]
and we construct a tensor by taking the outer product, \(t^{\prime} = \vec{u}^{\prime} \otimes \vec{v}^{\prime}\) then
\[t^{\prime} = u_i^{\prime} v_j^{\prime} = (a_{ik} u_k)(a_{jl} v_l) = a_{ik} a_{jl} u_k v_l = a_{ik} a_{jl} t_{kl}\]
and the inverse relation is then
\[t_{ij} = a_{kl} a_{lj} t^{\prime}_{kl}\]
This principle can be continued for higher and higher order tensors.
Physical examples
Example 9 (The moment of intertia tensor)
Rotational motion of a rigid body depends both on the axis and the moment of inertia with respect to the axis, with the moment of inertia, which, for a body composed of masses, \(\set{m_i}\) which are at a distance \(r_i\) from the axis can be described
\[I = \sum_i m_i r_i^2 = \int \rho r^2 \dd{V}\]
In order to have a means of calculating the moment of inertia along any axis we need a tensor.
In order to find the form of this tensor we turn to angular momentum.
The total angular momentum, \(\vec{J}\), of a body is the sum of all the angular momenta of its constituent parts,
\[\vec{J} = \sum_i \vec{L}_i = \sum_i \vec{r}_i \times m_i \vec{v}_i = \sum_i m_i \qty[ \vec{r}_i \times (\omega \times \vec{r}_i)]\]
We assume the body to be rigid, so that \(\vec{\omega}\) is constant for all its constituent particles.
Then
\[\begin{split}\begin{aligned}
\vec{J} & = \sum_i m_i \qty[ (\vec{r} \cdot \vec{r}) \vec{\omega} - (\vec{r}_i \cdot \vec{\omega} ) \vec{r}_i] \\
& = \sum_i m_i
\begin{pmatrix}
\omega_x (y_i^2 + z_i^2 ) & - \omega_y x_i y_i & - \omega_z x_i z_i \\
- \omega_x y_i x_i & \omega_y (z_i^2 + x_i^2) & - \omega_z y_i z_i \\
- \omega_x z_i x_i & - \omega_y z_i y_i & \omega_z (x_i^2 + y_i^2)
\end{pmatrix} \\
& =
\begin{pmatrix}
\sum_i m_i (y_i^2 + z_i^2) & - \sum_i m_i x_i y_i & - \sum_i m_i x_i z_i \\
- \sum_i m_i y_i x_i & \sum_i m_i (z_i^2 + x_i^2) & - \sum_i m_i y_i z_i \\
- \sum_i m_i z_i x_i & - \sum_i m_i z_i y_i & \sum_i m_i (x_i^2 + y_i^2)
\end{pmatrix}
\begin{pmatrix}
\omega_x \\
\omega_y \\
\omega_z
\end{pmatrix} \\ & = {\cal I} \vec{\omega}\end{aligned}\end{split}\]
Hence
Definition 70 (Moment of intertia tensor)
\[\begin{split}{\cal I} =
\begin{pmatrix}
\sum_i m_i (y_i^2 + z_i^2 ) & - \sum_i m_i x_i y_i & - \sum_i m_i x_i z_i \\
- \sum_i m_i y_i x_i & \sum_i m_i (z_i^2 + x_i^2) & - \sum_i m_i y_i z_i \\
- \sum_i m_i z_i x_i & - \sum_i m_i z_i y_i & \sum_i m_i (x_i^2 + y_i^2)
\end{pmatrix}\end{split}\]
The on-diagonal components of the moment of inertia tensor are the moments of inertia, while the off-diagonal elements are the products of inertia.
We can find the three moments of inertia,
\[\begin{split}\begin{aligned}
I_{xx} &= \sum m(y^2 + z^2) = ma^2 \qty( 4 \times \frac{3}{16} ) = \frac{3 ma^2}{4}\\
I_{yy} &= \sum m( x^2 + z^2 ) = ma^2 \qty( 2 \times\frac{9}{16} + 2 \times \frac{1}{16} ) = \frac{5 ma^2}{4} \\
I_{zz} &= \sum m(x^2 + y^2 ) = \frac{8 ma^2}{4}
\end{aligned}\end{split}\]
The products of inertia are straightforward, as \(z=0\) causes \(I_{xy} = I_{yz} = 0\), so
\[\begin{split}\begin{aligned}
I_{xy} &= - \sum m x y = - ma^2 \qty( 2 \times \frac{3}{4} \frac{\sqrt{3}}{4} - 2 \times \frac{1}{4} \frac{\sqrt{3}}{4} ) \\
&= - \frac{\sqrt{3}m a^2}{4}
\end{aligned}\end{split}\]
So
\[\begin{split}{\cal I} = \frac{ma^2}{4}
\begin{pmatrix}
3 & - \sqrt{3} & 0 \\
- \sqrt{3} & 5 & 0 \\
0 & 0 & 8
\end{pmatrix}\end{split}\]
By diagonalising the matrix we can find the principle moments of inertia.
From the characteristic equation of \({\cal I}\),
\[\begin{split}\begin{aligned}
\chi_{{\cal I}}(\mu) &= \qty( (3-\mu)(5-\mu)-3 ) = 0 \\ &= (2-\mu)(6-\mu) = 0\end{aligned}\end{split}\]
so \(\mu = \set{2,6,8}\). Thus
\[\begin{split}{\cal I} = \frac{ma^2}{4}
\begin{pmatrix}
2 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 8
\end{pmatrix}\end{split}\]
So the body’s principle moments of inertia are
\[I_{\rm prin} = \set{ \half ma^2, \frac{3}{2} ma^2, 2 ma^2}\]
Finally, to find the principle axes we need the corresponding eigenvectors to the eigenvalues, so \(\mu=2\),
\[\begin{split}\begin{aligned}
(3-2) x - \sqrt{3} y &= 0 \\
y &= \frac{1}{\sqrt{3}} x\end{aligned}\end{split}\]
\(\mu=8\),
\[\begin{split}\begin{aligned}
(3-8)x - \sqrt{3} y &= 0 \\
5x + \sqrt{3} y &= 0 \\
y &= \frac{-5}{\sqrt{3}} x\end{aligned}\end{split}\]
\(\mu=6\),
\[\begin{split}\begin{aligned}
(3-6) x - \sqrt{3} y &= 0 \\
y &= \frac{-3}{\sqrt{3}} x = - \sqrt{3} x\end{aligned}\end{split}\]
The Parallel Axis Theorem
Let \({\cal I}_G\) be the inertia tensor with respect to the centre of mass, \(G\) of a rigid body, and \({\cal I}_O\) be the tensor with respect to a different point \(O\) in the same coordinate frame.
Let \(\vec{r} = GO\).
From the definition of the moment of inertia tensor,
\[\begin{split}{\cal I}_O = {\cal I}_G + M
\begin{pmatrix}
r_y^2 + r_z^2 & -r_x r_y & - r_x r_z \\
- r_y r_x & r_z^2 + r_x^2 & - r_y r_z \\
- r_z r_x & - r_z r_y & r_x^2 + r_y^2
\end{pmatrix}\end{split}\]
which gives the parallel axis theorem.
Einstein Summation Convention
Consider a three-dimensional vector space, \(\mathsf{V}\), over the field of real numbers, \(\mathbb{R}\).
Any point in the space can be described by an ordered set of three numbers, \((x_1, x_2, x_3)\), known as coordinates, such that
\[\begin{split}\vec{A} = (x_1, x_2, x_3) \cdot
\begin{pmatrix} \vec{e_1} \\ \vec {e_2} \\ \vec{e_3} \end{pmatrix}\end{split}\]
where \(\vec{A}\) is any vector in \(\mathsf{V}\), and \(\vec{e_1}\), \(\vec{e_2}\), and \(\vec{e_3}\) constitute a basis for \(\mathsf{V}\).
This can be expressed in a more compact form by adopting the Einstein summation convention.
In this system the summation sign, \(\sum\), is omitted, and replaced with repeated indices, viz:
\[\vec{A} = A_i \vec{e_i} = \sum_{i=1}^3 A_i\vec{e_i}\]
here \(i\) is a repeated index, and so the summation over it is implicit.
This allows the definition of a number of vector calculus operations in a more compact way.