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Mathematics
Linear and Abstract Algebra
Orthogonal matrix construction
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[QUOTE="p4wp4w, post: 5940520, member: 584027"] Given a real-valued matrix ## \bar{B}_2=\begin{bmatrix} \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. [/QUOTE]
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Orthogonal matrix construction
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