when does $\mathbf{A}^{-1}\mathbf{B}$ and $\mathbf{T}\mathbf{B}\mathbf{T}$, where $\mathbf{A}^{-1}=\mathbf{T}^2$ have same eigenvalues. Let $\mathbf{A}$ and $\mathbf{B}$ be invertible matrices. Let $\mathbf{A}^{-1}=\mathbf{T}^2$. In that case, when does  $\mathbf{T}\mathbf{B}\mathbf{T}$ and $\mathbf{T}^2\mathbf{B}$ has same eigenvalues?

Since $T$ is invertible, and $T(TBT) T^{-1} = T^2B$, we see that $TBT$ and $T^2B$ are similar, hence they have the same eigenvalues.
