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In Griffiths E&M book, he says that a second rank tensor transforms with two factors of some transformational tensor on each of its nine components-I'm not sure why that is. I thought a second rank tensor was akin to a 2D matrix-which only has a transformation act on each component once.

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if tensors are defined over a 3 dimensional vector space then the dimension of the space of rank two tensors is nine – janmarqz Mar 8 '14 at 5:02

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You may know the transformation law of a matrix as a similarity transformation.

Suppose $M$ is a matrix acting according to a basis set $B$. Suppose $P$ is the matrix that transforms $B$ to a new basis $B'$. Then $M'$ is the matrix acting in the new basis, by

$$M' = P M P^{-1}$$

It may help to read off what's being done: for any input vector $v'$ in the new basis, $P^{-1}$ moves it to the original basis, $M$ acts on it, then $P$ takes the transformed vector back to the new basis.

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