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I'm looking for a matrix decomposition that deduce the "common" part of two matrices and their "remainders" relative each other.

More formally, given two invertible upper triangular matrices $A$ & $B$ from $\mathbb{R}^{n\times n}$ is there a matrix decomposition that compute three invertible upper triangular matrices $U, M_A$ and $M_B$ such that $A = U\times M_A$ and $B = U\times M_B$ where and $X \times Y$ represent the matrix multiplication of $X$ and $Y$?

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    You will need to give more details of what you actually want, otherwise you could just take $U=I$. – 2017-01-17
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    More generally, $U$ can be **any** invertible upper triangular matrix. – 2017-01-17

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