Page 157 - Textos de Matemática Vol. 44

THE MAIN DIAGONAL PRODUCTS OF 3 × 3 NORMAL MATRICES 147 2. 3 × 3 unistochastic matrices
It is efficient for the discussion of WkΠ(D) to introduce unistochastic matri- ces.
Recall that an n × n real matrix A = (aij ) is said to be doublystochastic if it is nonnegative and all its line (row or column) sums are equal to 1; it is orthostochastic if there is an O = (oij) ∈ O(n) so that A = (o2ij), and it is unistochastic if there is an U = (uij) ∈ U(n) so that A = (|uij|2). Instead of the orthogonal and unitary groups O(.),U(.) one can choose here just as well the special orthogonal or unitary groups SO(.) and SU(.). Let A(n) = {real n × n matrices with line sums 1}. This is a n2 − 2n + 1-dimensional affine space. Further let
D(n) = {doublystochastic n × n matrices},
Ort(n) = {orthostochastic n × n matrices},
Uni(n) = {unistochastic n × n matrices}.
For early work on the spaces A(n) and sets D(n), see [7], in particular p. 327, and [8]. In the sequel we work exclusively with the case n = 3.
We have strict inclusions A(3) ⊃ D(3) ⊃ Uni(3) ⊃ Ort(3). In fact 1 1 0
011
lies evidently in D(3) but not in Uni(3), while, by [11, p. 83],
2 1101
1 1 1 3
C0=1 1 1 1 111
lies in Uni(3) but evidently it lies not in Ort(3). So it is unfortunate that in parts of the older literature what we and many physicists nowadays call unistochastic is called orthostochastic.
Recall that a set S in an affine real space is starshaped with respect to s0 ∈ S if for all s ∈ S the segments s0s pertain to S. Equivalently, after translation of Sby−s0,wegetS−s0 =[0,1](S−s0)={λ(s−s0):0≤λ≤1,s∈S}.The boundary ∂ below is meant w.r.t. the affine hull of S − s0.
Lemma 2.1. Let S be a compact set that is starshaped with respect to s0. Then S−s0 =[0,1]∂(S−s0).
Proof. Clearly the set S − s0 is starshaped with respect to the origin (of the vectorspace in which the affine hull of S is considered). So we may assume s0 = 0, i.e. we assume S = [0,1]S. The inclusion S ⊇ [0,1]∂S is clear. Now