An $n\times n$ matrix $A$ with nonegative real entries $a_{ij}$ is said to be doubly stochastic if $\sum_{i=1}^na_{ij} = 1$, for all $j$, and $\sum_{j=1}^na_{ij}=1$, for all $i$.

Much is known [1] about the algebraic structure of the semigroup $\Omega _n$ formed by all doubly stochastic $n\times n$ matrices. For example, permutation matrices are the only invertible doubly stochastic matrices whose inverse is also doubly stochastic. On the other hand [3], the idempotent elements in $\Omega _n$ are precisely the direct sums of $k\times k$ matrices of the form $$ \pmatrix{ 1/k & 1/k & \ldots & 1/k \cr \vdots & \vdots & \ddots & \vdots\cr 1/k & 1/k & \ldots & 1/k \cr} $$ together with their conjugates by permutation matrices.

Question: Which doubly stochastic matrices are partial isometries (i.e. satisfy the equation $AA^tA = A$)?

See [2] for the characterization of normal, partial isometric, doubly stochastic matrices.

[1] Farahat, H. K., The semigroup of doubly-stochastic matrices, Proc. Glasg. Math. Assoc. 7, 178-183 (1966). ZBL0156.26001.

[2] Prasada Rao, P. S. S. N. V., On generalized inverses of doubly stochastic matrices, Sankhyā, Ser. A 35, 103-105 (1973). ZBL0301.15005.

[3] Sinkhorn, R., Two results concerning doubly stochastic matrices, Am. Math. Mon. 75, 632-634 (1968). ZBL0162.04205.