Transformation of tensor components

X.B.2.2 Transformation of tensor components

Similarly to what has been done for vectors in section X.B.2.1, one derives transformations of the components of tensors. One considers the components of tensor $T$ in two coordinate systems:

T = T_{k l} (e_{k} \otimes e_{l}) = T_{i j}^{'} (e_{i}^{'} \otimes e_{j}^{'}) .

Using the same definition of transformation matrix $A_{i j}$ as in section X.B.2.1, one writes:

e_{k} = A_{i k} e_{i}^{'},

e_{l} = A_{j l} e_{j}^{'} .

Then, the substitution of the two expressions in the equations defining components gives:

\begin{align} T = T_{k l} (e_{k} \otimes e_{l}) & = T_{i j}^{'} (e_{i}^{'} \otimes e_{j}^{'}), \\ T_{k l} (A_{i k} e_{i}^{'} \otimes A_{j l} e_{j}^{'}) & = T_{i j}^{'} (e_{i}^{'} \otimes e_{j}^{'}), \\ A_{i k} T_{k l} A_{j l} (e_{i}^{'} \otimes e_{j}^{'}) & = T_{i j}^{'} (e_{i}^{'} \otimes e_{j}^{'}) . \end{align}

The last expression allows us to extract a relation involving only the components of tensors, and not the base vectors:

T_{i j}^{'} = A_{i k} T_{k l} A_{j l} .

Here again, one recognizes a classical matricial expression:

T^{'} = A \cdot T \cdot A^{t} .

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