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\bar{B}_{21}\\

\bar{B}_{22}

\end{bmatrix}\in{R^{p \times m}}

##, I am looking for an orthogonal transformation matrix i.e., ##T^{-1}=T^T\in{R^{p \times p}}## that satisfies:

$$

\begin{bmatrix}

T_{11}^T & T_{21}^T\\

T_{12}^T & T_{22}^T

\end{bmatrix}\bar{B}_2=

\begin{bmatrix}

0\\

B_{2}

\end{bmatrix},

$$ where ##B_2\in{R^{m \times m}}## and non-singular. Assuming ##T=

\begin{bmatrix}

T_{11} & T_{12}\\

T_{21} & T_{22}

\end{bmatrix}##, and replacing the equation above, I can easily see ##T^{-1}## as

$$

\begin{bmatrix}

(T_{11}-T_{12}T_{22}^{-1}T_{21})^{-1} & -T_{11}^{-1}T_{12}(T_{22}-T_{21}T_{11}^{-1}T_{12})^{-1}\\

-T_{22}^{-1}T_{21}(T_{11}-T_{12}T_{22}^{-1}T_{21})^{-1} & (T_{22}-T_{21}T_{11}^{-1}T_{12})^{-1}

\end{bmatrix}=

\begin{bmatrix}

T_{11}^T & -T_{11}^T\bar{B}_{21}\bar{B}_{22}^{-1}\\

T_{12}^T & B_2\bar{B}_{22}^{-1}-T_{12}^T\bar{B}_{21}\bar{B}_{22}^{-1}

\end{bmatrix}.

$$ The literature says that there exists a transformation satisfying the last matrix equality. Accordingly, I wonder how can I synthesize ##T##. There should be a systematical method of obtaing ##T##, otherwise, I may carry out an exhaustive search.

As a background, in system theory, I am looking for a transformation that decomposes the system into the so-called regular form and ##\bar{B}_2## is a part of input matrix that defines the range space and causes the invariant RHS zeros.