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I know this question sounds lame, but the book I am following doesn't use the answer I expect and it has been using a similar notations everywhere else which has confused me.

I think Q[Any tensor] transpose Q can be written as Q(im)Q(mj), but the book has written Q(im)Q(jm) which, in my opinion, doesn't even satisfy the condition of matrix multiplication, so it should be wrong.

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$(Q\cdot Q)_{ij}=Q_{im}Q_{mj}$

$(Q^T\cdot Q)_{ij}=(Q^T)_{im}Q_{mj}=Q_{mi}Q_{mj}$

where we use that $(Q^T)_{im}=Q_{mi}$

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What you suggest is Q times Q, not Q transpose times Q. Check the definition of matrix multiplication.

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