I have a question about linear transformation and eigenvalues.
Given a linear transformation $T:R^3 \to R^3$, And: $E$ is the standard basis of $R^3$, $B$ is another basis of $R^3$.
Let's denote: $A=[T]^B_B$ .
Let's assume that after gaussian elimination process on $A$ we get a matrix $M$ with one row of $0$'s, and now we calculate the eigenvalues of $M$.
- Are the eigenvalues of $M$ also the eigenvalues of the transformation $T?$
I think yes, because the eigenvalues don't change when you change basis, but the correct answer is no, can someone explain to me why?
By the way, Is it correct to say that we always must to work only with the standard basis of $R^3$ to find the eigenvalues of $T?$ (i.e the eigenvalues of $T$ are the roots of the caracteristic polynomial $P_A = Det(A- \lambda \cdot I)$ where $A=[T]^E_E$ , and $E$ is the standard basis)?
Thanks for help!