# Index notation clarification

Previously, I have seen matrix notation of the form $T_{ij}$ and all the indices have been in the form of subscripts, such that $T_{ij}x_j$ implies contraction over $j$. However, recently I saw something of the form $T_i^j$ which seems to work not entirely differently from what I was used to. What is the difference? and how do they decide which index to write as a superscript and which a subscript? What is the point of writing them this way? Is there a difference?

(A link to a good reference explaining how these indices work would also be appreciated!)

Thanks.

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The staggered index notation $T^i{}_j$ works great in conjunction with the Einstein summation convention, where one of the rules is that an index that is summed over must appear once as a subscript and once as a superscript. Usually the index of an ordinary vector's components are written in superscript, so the contraction becomes $T^i{}_j x^j$.