Let $(V,\langle -,- \rangle _1), (W,\langle -,- \rangle _2)$ be finite-dimensional inner product spaces over a field $\mathbb{F}$ ($\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$)
Let $T:V\rightarrow W$ be a linear transformation with rank $r$.
Then, the singular values $\sigma_1≧...≧\sigma_r>0$ are uniquely determined.
Now, suppose another inner product is given to each vector spaces. Namely $\langle -,- \rangle_3$ and $\langle-,- \rangle_4$ for $V,W$ respectively.
Since inner product does not affect rank, singular values of $T$ can be determined as $\mu_1≧...≧\mu_r>0$.
My questions is that, are these two sequences the same?
That is $\mu_i = \sigma_i$?
I recommend looking at the mathematical preliminaries of this text, which go into a lot of detail about coordinate-free transformations with arbitrary inner products, and define singular values $\gamma_i$ for real transformations $A$ with respect to orthonormal bases $u_i, w_i$ by the values that, for all $x$, satisfy $$ Ax = \sum_{i=1}^r \gamma_i \langle u_i, x \rangle w_i $$