20211104 why are the traces of similar matrices the same

set up $\mathbf{A}={\left(a_{ij}\right)}_{m\times n},\mathbf{B}={\left(b_{ij}\right)}_{n\times m}$, be $\mathrm{tr}(\mathbf{A}\mathbf{B})=\mathrm{tr}(\mathbf{B}\mathbf{A})$;
Then we know $\mathbf{P}$ It's a nonsingular matrix
$$\mathrm{tr}(\mathbf{B})=\mathrm{tr}\left({\mathbf{P}}^{-1}\mathbf{A}\mathbf{P}\right)=\mathrm{tr}\left({\mathbf{A}\mathbf{P}\mathbf{P}}^{-1}\right)=\mathrm{tr}(\mathbf{A})$$

One more sentence ,$\mathbf{A}\mathbf{B}$ and $\mathbf{B}\mathbf{A}$ The nonzero eigenvalues of are the same , The demonstration process is as follows
https://blog.csdn.net/Europe233/article/details/86727078

in addition , The eigenvalues of similar matrices are the same

https://www.zhihu.com/question/437714477/answer/1658995521