Tensors and Their Transformations

This post introduces the definition of a Cartesian tensor and the intuition behind it, then works through a change of basis in an example.

Definition

Consider a rotation of the basis \(\{\mathbf{e}_i\}\) (frame \(S\)) to a new basis \(\{\mathbf{e}'_i\}\) (frame \(S'\)). This is a passive rotation: the vectors stay put and the axes move. Define the transformation matrix

\[\ell_{ij} = \mathbf{e}'_i \cdot \mathbf{e}_j .\]

The components of any vectors \(\mathbf{a}\) and \(\mathbf{b}\) in the two frames are related by

\[a'_i = \ell_{ij}\, a_j, \qquad b'_i = \ell_{ij}\, b_j,\]

with summation over repeated indices. A tensor of rank \(n\) is a set of components that transform in the same way in every index:

\[T'_{ij\cdots k} = \ell_{ip}\, \ell_{jq} \cdots \ell_{kr}\, T_{pq\cdots r}.\]

For a rank-2 tensor this is

\[T'_{ij} = \ell_{ip}\, \ell_{jq}\, T_{pq} = \ell_{ip}\, T_{pq}\, (L^{T})_{qj},\]

which in matrix form reads

\[T'=L\,T\,L^{T},\]

where \(L\) is the matrix with entries \(\ell_{ij}\).

Another way to see the transformation law

A rank-2 tensor \(T\) can be thought of as a linear map that sends a vector \(\mathbf{v}\) to \(T\mathbf{v}\). In the new frame, the same map sends \(\mathbf{v}'\) to \(T'\mathbf{v}'\).

To compute \(T\mathbf{v}\) we may instead:

1. rotate into the new basis with \(L\);
2. apply \(T'\);
3. rotate back with \(L^{-1}\).

This gives \(T\mathbf{v}=L^{-1}T'L\,\mathbf{v}\) for every \(\mathbf{v}\), so \(T'=L\,T\,L^{-1}\). Rotation matrices are orthogonal, \(L^{-1}=L^{T}\), so this agrees with the index calculation above.

Example: an inertia tensor

The principal moments of inertia of a rigid body about the axes \(\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\) are \(1\), \(2\) and \(3\). An observer uses the basis

\[\mathbf{e}_1'=\tfrac{1}{\sqrt{2}}\left(\mathbf{e}_1+ \mathbf{e}_2\right), \qquad \mathbf{e}_2'=\tfrac{1}{\sqrt{2}}\left(-\mathbf{e}_1+ \mathbf{e}_2\right), \qquad \mathbf{e}_3'=\mathbf{e}_3 .\]

What components of the inertia tensor does the observer measure?

Solution

From \(\ell_{ij}=\mathbf{e}_i' \cdot \mathbf{e}_j\),

\[L=\frac{1}{\sqrt 2}\begin{pmatrix} 1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & \sqrt{2} \end{pmatrix}, \qquad I=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} .\]

It is tempting to transform the inertia tensor like a vector, but it is a rank-2 tensor, so the transformation law applies to each index:

\[I_{i j}^{\prime} =\ell_{i p}\, \ell_{j q}\, I_{p q} =\left(L\, I\, L^T\right)_{i j}.\]

Multiplying out,

\[I' = \frac{1}{2}\begin{pmatrix} 1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & \sqrt{2} \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} \begin{pmatrix} 1 & -1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & \sqrt{2} \end{pmatrix} =\begin{pmatrix} \frac{3}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & \frac{3}{2} & 0 \\ 0 & 0 & 3 \end{pmatrix} .\]

As a check, \(I'\) is still symmetric, as an inertia tensor must be in any frame, and its trace (the sum of the principal moments, \(6\)) is unchanged.

Exercises

• What is the rank of a vector? Of a scalar? (Undergraduate, year 2)
• Show that \(\det L = +1\) for rotations and \(\det L = -1\) for reflections and inversions of the basis. (Undergraduate, years 1–2)
• Let \(\mathbf{c} = \mathbf{a} \times \mathbf{b}\). Show that \(c'_i = (\det L)\, \ell_{ij}\, c_j\) by considering rotations, reflections and inversions. Objects that transform like this are called pseudovectors. (Undergraduate, year 2)
• Give an example of a pseudovector in physics. (Undergraduate, year 2)
• () Define the Levi-Civita symbol and show that it is a rank-3 pseudotensor. (Undergraduate, years 2–3*)
• (*) Give an example of a pseudoscalar in physics. (Hint: helicity.) (Undergraduate, years 3–4*)

Credits

The intuition section is inspired by this Math Stack Exchange answer. The definition of tensors follows Dr B. J. Pendleton’s lecture notes for Electromagnetism and Relativity at the University of Edinburgh, and the example is adapted from one of his problem sheets.

Changes on 25 October 2026

What was wrong.

• The transformation of a rank-2 tensor was written \(T' = LTL\). It is \(T' = LTL^{\mathsf T}\).
• The worked example had \(-\tfrac12\) where the answer is \(+\tfrac12\).
• The rule for pseudovectors, \(\mathbf{c}' = \det L\, \mathbf{c}\), left out the rotation itself. It is \(c'_i = (\det L)\,\ell_{ij} c_j\).
• Two links to course notes no longer worked.

What changed. All are fixed, the dead links are removed and the index notation is consistent. Previously titled “Tensors and its transformations”.