I have very little experience and exposure to working with tensors. I know what a tensor is definitionally from a linear algebra course, and I know that rank 2 tensors transform in a particular way (this is the extent of my knowledge).
My question: Is there a way to determine if two tensors are the same even if their components are different?
This questions comes about because I am using mathematica to convert a 2-rank tensor from spherical coordinates to cartesian coordinates. I then converted the tensor back into spherical coordinates. The matrix I ended up with is different that the one I stated with. My guess is that this is happening because the rank 2 tensor transform multiplies one side by a matrix and the other by that matrix's transpose. Thus, performing this transformation and then performing the transform in the opposite way DOES NOT mean you're net multiplying the original tensor by identity matrices. So, perhaps you do not end up with what you started with.
Attached is a picture of my initial and end matrix (in spherical coordinates). All help would be greatly appreciated!
Edit: Added code used in case that is helpful.
Best Answer
It is important to distinguish between a linear transformation (which you call a matrix) and a $2$-tensor. From the definition of the quadrupole moment, it appears that it is a $2$-tensor.
Recall that, given a vector space $V$, a $2$-tensor is a function $\newcommand\R{\mathbb{R}}$ $$ T: V\times V \rightarrow \R $$ that is bilinear. If you choose a basis $(b_1,b_2, b_3)$ of $V$, then the tensor is uniquely determined by its value for pairs of basis vectors, $$ T_{ij} = T(b_i,b_j). $$ If you remember this, then it is straightforward to figure out how the components change when you change to a new basis of $V$. But observe that the components of $T$ are indeed uniquely determined by $T$ itself and the basis $(b_1,b_2,b_3)$. So it's not possible for two tensors to be equal if their components are not.
Now observe that if you have a new basis $(e_1, e_2, e_3)$, then you can write the old basis in terms of a new basis, using a matrix, \begin{align*} \begin{bmatrix} b_1 & b_2 & b_3\end{bmatrix} &= \begin{bmatrix} e_1 & e_2 & e_3 \end{bmatrix} \begin{bmatrix} M^1_1 & M^1_2 & M^1_3 \\ M^2_1 & M^2_2 & M^2_3 \\ M^3_1 & M^3_2 & M^3_3 \end{bmatrix} \end{align*} The components of the tensor with respect to the new basis are now easily computed using the bilinear property of $T$: $$ T(e_i, e_j) = T(b_kM^k_i, b_lM^l_j) = M^k_iM^l_jT(b_k,b_l). $$ If you call the components of the tensor with respect to the new basis $\widehat{T}_{ij}$, then the formula is $$ \widehat{T}_{ij} = M^k_iM^l_jT_{kl}. $$ Although it is easy to remember this formula, it is always hard for me to remember whether the components on the left are with respect to the old or new basis. So I just remember the abstract definitions and work out the formula from scratch each time I need it.
I think it is indeed a good idea to do this computation by hand first.