MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given a unitary matrix $A$ with entries $a_{ii}$, it's clear that the matrix $B$ with entries $b_{ii} = |a_{ii}|^2$ is doubly stochastic. Is the inverse of this statement true? Namely, given a doubly stochastic matrix $B'$ with entries $\beta_{ii}$, does there exist a unitary matrix with entries $\alpha_{ii}$ such that $|\alpha|^2_{ii} = \beta_{ii}$?

share|cite|improve this question
up vote 2 down vote accepted

A brief googling yielded another interesting paper:

Topological properties of orthostochastic matrices ☆

Tony F. Heinz

"In this paper, it is shown that for n⩾3 the orthostochastic matrices are not everywhere dense in the set of doubly stochastic matrices, thus answering a question of L. Mirsky in his survey article on doubly stochastic matrices [2]"

share|cite|improve this answer
That answers the question directly - thank you! – Ben Lerner Mar 29 '12 at 1:25

Counterexample: $$\begin{pmatrix} 0 & 1/2 & 1/2 \\ 1/2 & 0 & 1/2 \\ 1/2 & 1/2 & 0 \end{pmatrix}.$$ The question is of interest in quantum probability. Your map from unitary matrices to doubly stochastic matrices defines an interesting region which has non-zero volume, and which cannot be convex because it visits all of the vertices.

share|cite|improve this answer
Stochasm is the lowest form of wit. – Will Jagy Nov 7 '11 at 2:53
@WillJagy Your comment still brings out a belly laugh four years later. – WetSavannaAnimal aka Rod Vance Jan 26 '15 at 12:41

Interesting topic. Today we do not have a clear picture about the relationship between being uni-stochastic (or ortho-stochastic, if you restrict your attention to orthogonal matrices) and doubly-stochastic.

A couple of references:

Defect of a unitary matrix, Wojciech Tadej, Karol Zyczkowski

Recent work, it contains a number of references on the discussion.

On the digraph of a unitary matrix, Simone Severini

A more combinatorial perspective (but superficial)

share|cite|improve this answer

Yet another reference

I. Bengtsson, A. Ericsson, M. Kus, W. Tadej, and K. Zyczkowski, Birkhoff's polytope and unistochastic matrices, N=3 and N=4, Comm. Math. Phys. 259, 307-324 (2005).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.