Inverse of doubly stochastic matrix M is doubly stochastic iff M is a permution matrix

manpreet Best Answer 2 years ago

In a very short remark, my syllabus states that if M∈ΩnM∈Ωn (i.e. the set of doubly stochastic matrices) and MM is invertible, that M−1∈ΩnM−1∈Ωn if and only if MM is a permutation matrix. I'm having trouble to see why this is so obvious.

By definition of a doubly stochastic matrix, Me=eMe=e and eTM=eTeTM=eT, and thus M−1Me=M−1e=eM−1Me=M−1e=e, and similarly eTM−1=eTeTM−1=eT. But of course, we do not know that all matrix elements are non-negative, and by trying a few examples it indeed turns out that (for all matrices I tried), doubly stochastic matrices which are not permutation matrices do not have non-negative inverses.

For the given remark, I'm trying to prove the equivalence. The reverse implication is easy: SnSn is a group, and h:margin: 0px; padding: 0px; border: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 16.65px; vertical-align: 0px; box-sizing: content-box; transition: none 0s ease 0s; position: absolute; clip: rect(3.126em, 1000.66em, 4.207em, -