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.