Principal angles between random subspaces and polynomials in two free projections

We use the geometric concept of principal angles between subspaces to compute the noncommutative distribution of an expression involving two free projections. For example, this allows to simplify a formula by Fevrier–Mastnak–Nica–Szpojankowski about the free Bernoulli anticommutator. We also derive economically an explicit formula for the free additive convolution of Bernoulli distributions. As a byproduct, we observe the remarkable fact that the principal angles between random half-dimensional subspaces are asymptotically distributed according to the uniform measure on [0, π/2]. 1. Principal angles Let K be the real or complex field. For a integer n, we equip Kn with its usual inner product. We set [n] = {1, . , n}. For 0 6 k 6 n, we denote by Gn,k the Grassmann manifold defined as the set of all k-dimensional subspaces ofKn. Given a subspace E ⊂ Kn, we denote by PE the orthogonal projection onto E. We now introduce the concept of principal angles which play a central role in this note. Principal angles between two subspaces generalize the notion of the angle between two lines in K2. They are defined through the following proposition. Proposition 1. Let 0 6 k, l 6 n and consider subspaces E ∈ Gn,k and F ∈ Gn,l. There exist (1) an orthonormal basis (ei)i∈[k] of E, (2) an orthonormal basis (fj)j∈[l] of F , (3) numbers θ1 6 θ2 6 · · · 6 θmin(k,l) in [0, π/2] such that, for every i ∈ [k] and j ∈ [l] 〈ei, fj〉 = { 0 if i 6= j cos(θi) if i = j. Moreover, the numbers (θi)i∈[min(k,l)] are uniquely defined by these conditions. In the context of Proposition 1, the numbers (θi)i∈[min(k,l)] are called the principal angles between E and F . The vectors ei and fj are sometimes called the principal vectors; they are not uniquely defined. 1 ar X iv :2 10 9. 06 53 5v 1 [ m at h. PR ] 1 4 Se p 20 21 Principal angles are discussed in several places (see, e.g., [2, 4]) and can be related to singular values. If ei, fj and θi satisfy the condition of Proposition 1, then PEPF = ∑ i∈[k] |ei〉〈ei| ∑ j∈[l] |fj〉〈fj |  = ∑ i∈[min(k,l)] cos(θi)|ei〉〈fi| is a singular value decomposition of the operator PEPF . Conversely, one may prove Proposition 1 by considering a singular value decomposition of PEPF ; the uniqueness of principal angles follows from the uniqueness of singular values. We compute, on few simple examples, the spectrum of a self-adjoint expression in two orthogonal projections from the principal angles between their ranges. Proposition 2. Let E ∈ Gn,k and F ∈ Gn,l with k 6 l. Let m = dim(E∩F ) and (θi)i∈[k−m] the nonzero principal angles between E and F . Set P = PE and Q = PF . Then (1) the spectrum of PQP or QPQ is σ(PQP ) = σ(QPQ) = {0(n−k)} ∪ {cos θi} ∪ {1(m)}, (2) the spectrum of P +Q is σ(P +Q) = {0(n−k−l+m)} ∪ {1− cos θi} ∪ {1(l−k)} ∪ {1 + cos θi} ∪ {2(m)}, (3) the spectrum of ı(PQ−QP ) is σ(ı(PQ−QP )) = {− cos θi sin θi} ∪ {0(n−2k+2m)} ∪ {cos θi sin θi}, (4) the spectrum of PQ+QP is σ(PQ+QP ) = {cos θi − cos θi} ∪ {0(n−2k+m)} ∪ {cos θi + cos θi} ∪ {2(m)}. In these formulas, the spectrum is counted with multiplicity, the index i ranges in [k −m] and the notation λ(p) stands for the eigenvalue λ repeated p times. More generally, the spectrum of any self-adjoint polynomial in PE , PF depends only on the principal angles between E and F . Proof. Let (ei)i∈[k] and (fj)j∈[l] be respective orthonormal bases of E and F satisfying the conclusion of Proposition 1. We have ei = fi for i ∈ [m]. Consider the orthogonal direct sum