What all possible matrix reprsentations a linear operator can have? $L$ is a linear operator such that $L:V \to V$ where $V$ is a $n$ dimensional hilbert space. If $[L]_{ij}$ is the matrix representation for $L$ in the input and output basis $\{i\}$ and $\{j\}$, then what is common in all the matrices $[L]_{ij}$  for varying $\{i\}$ and $\{j\}$. I feel they all have the same rank ( if I am not wrong ) but can't prove it definitely.
 A: Say that the matrix is $A_L$ in some basis (e.g. the standard basis) $\{e_i\}$. Say your input basis is $\{f_i\}$ and your output basis is $\{d_i\}$. Let the base change matrix from $\{e_i\}$ to $\{f_i\}$ be $T_f$ and the base change matrix from $\{e_i\}$ to $\{d_i\}$ be $T_d$, then you can do the following, given a vector $x_f$ in your input basis:


*

*Convert $x_f$ from $\{f_i\}$ to $\{e_i\}$: $x_e = T_f^{-1} x_f$.

*Apply $A_L$: $y_e = A_Lx_e = A_LT_f^{-1}x_f$.

*Convert $y_e$ from $\{e_i\}$ to $\{d_i\}$: $y_d = T_dy_e = T_dA_LT_f^{-1}x_f$.


I.e. you have
$$[L]_{f,d} = T_dA_LT_f^{-1}$$
since $T_d$ and $T_f$ both have full rank, we immediately get
$$\operatorname{rank} [L]_{f,d} = \operatorname{rank} A_L$$
from well-known properties of rank, i.e. the rank is independent of $f$ and $d$.
Other properties, e.g. the determinant will not be independent of $f$ and $d$:
$$\det [L]_{f,d} = \det T_d \det A_L \frac{1}{\det T_f}$$
but if you constrain $f$ and $d$ to be orthonormal bases, it will be the same up to a sign difference.
This question is basically asking what properties of linear transformations are invariant under the action of the product of the general linear group, $GL(V) \times GL(V)$ defined by
$$(g_1, g_2) \cdot L = g_1 L g_2^{-1}.$$
