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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!)


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up vote 3 down vote accepted

Mostly it's just a matter of the author's preference.

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 $T^i{}_j x^j$.

This rule can become relevant when one is working with multiple bases, in which cases supscript and superscript indices behave differently under basis change. Writing the matrix with staggered indices then servers a reminder that you're planning to use the matrix to represent a linear transformation, rather than to represent a bilinear form, for which both indices are always on the same level. This agrees with the fact that the matrix of a linear transformation and the matrix of a bilinear form respond differently to basis changes.

These considerations are most weighty in contexts where one needs to juggle a lot of basis changes -- or just to be sure that what one is writing does not depend on the particular choice of basis -- such as differential geometry. On the other hand, in introductory texts where this is less of an issue, there's an argument that explaining the rules for different kind of indices will just confuse the student without really adding to his understanding (as I have may confused you in the above paragraph).

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Thank you for the clear explanation! – striker Jun 10 '12 at 23:33

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