I want to transform a linear system matrix from x basis to z basis where Z=TrX. Tr is obtained through QR factorization. The linear system is in state-space form. Furthermore, I can use the Transformation to obtain regular form matrix for Eigenvalue placement design or Linear-Quadratic minimization feedback sliding mode control.
What I understand from the question is if A is the matrix of your linear system in X basis then one can always get its QR factorization and find its eigenvalues through QR algorithm. Now write your system as per your eigenvalues obtained.
You can easily compute or analyze rectangular matrices via SVD, Schur, or QR etc ... IN MATLAB:
https://au.mathworks.com/help/matlab/ref/double.svd.html (SVD)
https://au.mathworks.com/help/matlab/ref/schur.html (Schur)
https://au.mathworks.com/help/matlab/ref/qr.html (QR)
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