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I am going through 'INTRODUCTION TO TENSOR ANALYSIS' by myself, and there is something I quite don't get it. Thank you in advance for the answer.

Let $A$ be a tensor. A trace of $A$ is defined as

$tr(A)=A:I$

where a double dot product between dyad $a\otimes b$ and $c\otimes d$ is defined as $(a\otimes b):(c\otimes d)=(a\cdot c)(b\cdot d)$. Using basis vector notation, I lead

$tr({A}\cdot {B^T})={A}_{ij}{B}_{ij}$.

And I expected

$tr{A^2}=A^2:I =A_{ik}A_{kj}e_i\otimes e_j:e_l\otimes e_l =A_{ik}A_{kj}\delta_{il}\delta_{jl} =A_{lk}A_{kl}(=A_{ij}A_{ji})$,

not

$tr{A^2}=A_{ij}A_{ij}.$

which is written in the book. I have tried to figure out but, I have no idea what I missed.

This is my first time writing my question on this page, please let me know if there are something hard to understand.

lucy
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  • Assuming the linear algebra matrix multiplication $A^2=A_{ij}A_{jk}$ I get the same trace as you. Does your book assume $A^2$ like this or is $A$ symmetric ? – Kurt G. Aug 13 '23 at 16:22
  • @KurtG. No it didn't say $A$ is symmetric. Also it uses this formula on next page so I am not sure if it could be typo or something. – lucy Aug 13 '23 at 17:35
  • Could be (a typo). Read on and keep it in mind. – Kurt G. Aug 13 '23 at 17:44
  • Per your definition of the double dot product, $e_i\otimes e_j:e_l\otimes e_l = \delta_{ij}\delta_{ll}$, not $\delta_{il}\delta_{jl}$. – Paul Sinclair Aug 14 '23 at 19:40
  • @PaulSinclair sorry for the typo from the definition. I just fixed it – lucy Aug 16 '23 at 13:39

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