I have a reference $N \times N$ symmetric matrix -- with distinct eigenvalues -- decomposed using SVD as: $$ R_{ref} = V_{ref} D_{ref} V^{-1}_{ref} $$
If i get a matrix $S$ -- with distinct eigenvalues -- obtained from a similar physical process (i.e. defined in the same basis as $R_{ref}$), and also decomposes it with SVD such as: $$ S = W E W^{-1} $$
Then how to interpret the "projection" $T$ of $S$ on the eigenspace $V_{ref}$ defined by: $$ T = V^{-1}_{ref} S V_{ref} = V^{-1}_{ref} W E W^{-1} V_{ref} = (W^{-1} V_{ref})^{-1} E (W^{-1} V_{ref})$$
I tend to interpret $T$ as a measure of overlap between the eigenspace defined by $V_{ref}$ and the eigenspace defined by $W$. I say "the eigenspace" because the eigenvectors are orthonormal and associated to distinct eigenvalues but I am still not sure this is right to assume uniqueness.
Going back to interpretation. If for example $S=R_{ref}$ then using the same SVD decomposition, I should get $W=V_{ref}$ and $E = D_{ref}$, thus $T = E = D_{ref}$. In this case there is "perfect overlap" and we get the eigenvalues of $S$.
But what about a different matrix $S$? From the formula $T= (W^{-1} V_{ref})^{-1} E (W^{-1} V_{ref})$ I can think of $W^{-1} V_{ref}$ representing a rotation, that, if equal to identity, provides the "maximum overlap". But what if this rotation is not identity? I feel a diagonal elements $t_{ii}$ of $T$ represents the overlap (cross-product) between the eigenvector $v_i$ of the reference eigenspace of $R_{ref}$ and the eigenvector $w_i$ of the eigenspace of $S$, with $t_{ii}$ which could be small due to misalignment of the eigenspaces and/or a lower eigenvalue $e_{ii}$ of $S$.
However this interpretation is still confusing for me. In particular, how to guarantee that an SVD decomposition would always provide the eigenvectors of $W$ in the right order? In a typical implementation they are provided in order of increasing/decreasing eigenvalues and I feel that could result in a different order for slightly different matrix $S$ and therefore in totally different rotation $W^{-1} V_{ref}$, and consequently in a different $T$. Paradoxically the formula $ T = V^{-1}_{ref} S V_{ref} $ intuitively would not provide a largely different $T$ for slightly different $S$.
My apologies for the confusing post, but the question is precisely about clarifying the correct interpretation.