Matrix representation of linear transformation in different basis Given a linear transformation $$T:V\rightarrow V$$
And given 2 bases $B$ and $C$ of the subspace $V$

Is there any connection between $[T]_B$ and $[T]_C$ (the matrix representation of the transformation in the specified basis)

Does those 2 matrices have the same rank?signature?eigenvalues?
Are they congruence or similar?

Thanks :)

Update:

I'm leaning on thinking that they are congruence but no similar and there for have same rank and signature but not same eigenvaluess
Use that if $T_{B}$ and $T_{C}$ are matrices of the same linear map (or restriction of this map on some subspace of $V$), then exist the matrix $S:T_{B}=ST_{C}S^{-1}$
The expression you're looking for is $$B \circ [T]_B \circ B^{-1} = C \circ [T]_C \circ C^{-1}.$$ Let me explain.

The way I like to think of bases is:


  Definition 0. A basis of $V$ is an isomorphism $\varphi : V \leftarrow \mathbb{R}^{\mathrm{dim}(V)}$.


Now suppose we're given:


A vectorspace $V$
A natural number $n$ defined by $n = \mathrm{dim}(V)$
A linear transform $T : V \leftarrow V$.
Two bases $B$ and $C$ of $V$.


Then we get "matrices" $$B^{-1}\circ T \circ B : \mathbb{R}^n \leftarrow \mathbb{R}^n, \qquad C^{-1} \circ T \circ C : \mathbb{R}^n \leftarrow \mathbb{R}^n$$

which are really just linear transforms. So define:

$$[T]_B = B^{-1} \circ T \circ B, \qquad [T]_C = C^{-1} \circ T \circ C$$

Then obviously, these are related by:

$$B \circ [T]_B \circ B^{-1} = C \circ [T]_C \circ C^{-1}$$