I am searching for a way to represent differentiation of polynomials without losing information - and some rule to minimize the risk to using information that is not available. What I have mostly used so far is the non-invertible:
$${\bf D} = \left[\begin{array}{ccccc}0&1&0&0&0\\0&0&2&0&0\\0&0&0&3&0\\0&0&0&0&4\\0&0&0&0&0\end{array}\right]$$
We can build for example a Moore-Penrose inverse to find an integration matrix, which gives us:
$${\bf S = D}^{+} = \left[\begin{array}{ccccc}0&0&0&0&0\\1&0&0&0&0\\0&\frac 1 2 &0&0&0\\0&0&\frac 1 3&0&0\\0&0&0&\frac 1 4&0\end{array}\right]$$
This gives us ${(\bf SD})_{ij} = \cases{1,i=j\neq 1\\0,i=j=1\\0,i\neq j}, ({\bf DS})_{ij} = \cases{1,i=j\neq n\\0,i=j=n\\0,i\neq j}$, which is almost a unit matrix except the first or last coefficient is "lost" (depending on the order of operations). How can we fix this so that information is stored without risking to introduce irreversible confusion in our calculations?