In control theory, the Kalman decomposition is used to decompose a system, so the observable and controllable states can be distinguished between the unobservable and uncontrollable states.
Consider the following discrete time system \begin{align} x_{k+1} &= Ax_k + Bu_k \\ y_k &= Cx_k + Du_k \end{align}
The Kalman decomposition is given by $\hat{A} = TAT^{-1}$, $\hat{B}=TB$, $\hat{C}=CT^{-1}$, and $\hat{D} = D$ where $T^{-1}$ is an $n \times n$ invertable matrix. The resulting matrices $\hat{A}$, $\hat{B}$, $\hat{C}$, and $\hat{D}$ have a block structure corresponding to the controllable, observable, uncontrollable, and unobservable states. Now, my question is the following:
Is the transpose of the similarity transform equal to the inverse $T^{-1} = T^T$?
I have looked through several control theory books, and the decomposition is written as described previous, but I have seen from several sources that the decomposition is written with $T^T$ instead of $T^{-1}$. For example, see the function in MATLAB for performing the Kalman decomposition. The resulting similarity transform uses a transpose instead of an inverse. Similarly, the book "Modern Control Theory" by Brogan uses a transpose instead of an inverse as well.