Corank-1 projections and the randomised Horn problem

Harmonic analysis for rank-1 Randomised Horn Problems

“…A = U aU † and B = V bV † with " †" the Hermitian adjunct, are drawn from the Haar measure µ(dU ) of the unitary group U(n). Horn's question can then be rephrased to an explicit expression for the joint eigenvalue probability density of the eigenvalues c. There are general discussions on this particular randomised Horn problem [49] as well as specialisations to a rank 1 matrix b, see [20,12,17]. In the latter case, a closed analytic expression of the joint eigenvalue density is accessible; see also (4).…”

“…In [49,17], an harmonic analysis approach via the Fourier transform has been suggested. It works along the same ideas of characteristic functions in probability theory, where the density of the sum of independent random variables can be obtained by taking the inverse transform of the product of their corresponding characteristic functions.…”

“…With this analytical tool in mind, one can address a multiplicative versions of Horn's problem. One version of such a question is the projection of a Hermitian matrix A to a co-rank 1 hyperplane, see [2,13,17], where anew an explicit closed expression of the joint eigenvalue density can be derived. We will study a related problem where A and B are either positive definite Hermitian matrices and the product matrix is C = A 1/2 BA 1/2 or are unitary with C = AB.…”

“…1. (Sum on Herm(n), see [20,17,12]) The eigenvalues c of C = U aU † + V bV † are distributed according to the joint density…”

“…Specifically, in relation to the latter case, we are not aware that the transform has been defined in literature, although it is rather straightforward due to its intimate relation to the character expansion of square-integrable functions on U(n). The proof for the sum of two matrices is known [20,17,12] and shall serve as an illustration for the main steps of the proof for the other two cases. The proof of the known result ( 6) is an alternative one given in [17] where the Harish-Chandra-Itzykson-Zuber integral [24,28] has been employed.…”