Definition....
Suppose you have a 2 contravariant tensor
$$ T=T^{ij} e_i \otimes e_j $$in the canonical basis $\{e_i\}$ for $\mathbb{R}^3$. If we apply a basis change given by the matrix $M$ (that is to say, the observer moves his point of view), the vectors change its coordinates $(v^1,v^2,v^3)=v^i e_i$ to
$$ N \cdot \left(\begin{array}{c} v^1 \\ v^2\\ v^3 \end{array}\right) $$where $N=M^{-1}$.
But, what about our 2-tensor $T$?
Doing the maths you can conclude that if we express the coordinates $T^{ij}$ in a matrix, the new matrix is
$$ N T N^T $$In index notation this can be written
$$ N_{j}^a N^b_{i} T^{ij} $$This formula works in any rank other that 2.
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Author of the notes: Antonio J. Pan-Collantes
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