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For questions about factorization, the decomposition of mathematical objects (e.g. natural numbers, polynomials) into products of smaller objects (e.g. primes, lower degree polynomials).

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Finding a particular matrix factor

An example of such a factorization is given, for instance, by $$ C(x)=\begin{bmatrix} 1 & 0 \\ 0 & x\end{bmatrix}. $$ My question. … Does there exist a factorization of $A(x)$ as in \eqref{fact} such that the factor $C(x)$ possesses the two additional properties below? …