So I am inspired by unitary matrices which preserve the L2-norm of all vectors, so in particular the unit norm vectors. But then I saw that the L1-norm of probability vectors is preserved by matrices whose columns are probability vectors. And this got me thinking: But what are the matrices preserving the L1-norm of arbitrary real unit L1-norm vectors? So basically we extend a probability vector to also allow a sign, but ignoring the signs, this should still be a probability vector; and then we ask for the corresponding structure-preserving matrices.

It is already clear that the columns of such a matrix should be this ‘extended’ kind of probability vector, because we can multiply the matrix with a standard basis vector which has L1-norm 1. But not all of such matrices preserve this, take for example $ M = \frac{1}{2} \left(\begin{matrix} 1 & 1\ 1 & -1 \end{matrix}\right) $ and $ x = \left( \begin{matrix} 0.3 \ -0.7 \end{matrix} \right) $ . Then we have $ Mx = \left(\begin{matrix} -0.2 \ 0.5 \end{matrix}\right) $ which fails the test.