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Corank-1 projections and the randomised Horn problem. (English) Zbl 1448.15045

Let \(\hat{x}\) be a random \(n\times 1\) vector uniformly distributed on the unit complex sphere. Consider the matrix \(\Pi= \mathrm{I}_{n} - \hat{x} \hat{x}^{†}\) (the corank-1 projection on the hyperplane orthogonal to \(\hat{x}\)). Let \(A\) be an \(n\times n\) complex Hermitian matrix with eigenvalues \(a_{1}>a_{2} > \dots >a_{n}.\) For the random matrix \(B= \Pi A \Pi\), it is a well known fact (see [P. J. Forrester and E. M. Rains, Probab. Theory Relat. Fields 131, No. 1, 1–61 (2005; Zbl 1056.05142)]) that it has one zero eigenvalue and non-zero eigenvalues \(\lambda_{k}\), \(k=1, 2, \dots, n-1,\) interlacing with those of \(A.\) On the other hand, the probability density function (PDF, in short) is given by \(\Gamma(n) \frac{ \prod _{1\leq j <k \leq n-1} (\lambda_{j}- \lambda_{k})}{ \prod _{1\leq j <k \leq n} (a_{j}- a_{k})}.\) On the other hand, for \(A\) and \(\hat{x}\) as above, consider the random matrix \(C=A+ b \hat{x} \hat{x}^{†}\) with \(b\) a positive real parameter. In [J. Faraut, Tunis. J. Math. 1, No. 4, 585–606 (2019; Zbl 1406.15012)], it is proved that the structure of its eigenvalues in terms of those of \(A\) as well as its probability density function follow the same patterns as in the previous case.
In this work, the authors give a unified derivation of both results showing that they are special cases of the random matrix sum \(UAU^{†}+ VBV^{†},\) where \(A,B\) are fixed Hermitian matrices and \(U, V\in U(n)\) are chosen with Haar measure. If \(A\) and \(B\) are real symmetric and \(U, V\in \mathrm{O}(n),\) the case when \(B\) has rank 1 is analyzed. A randomised multiplicative form of Horn’s problem involving unitary matrices is studied. Furthermore, if \(A\) and \(B\) have full rank, using a result in [J.-B. Zuber, Ann. Inst. Henri Poincaré D, Comb. Phys. Interact. (AIHPD) 5, No. 3, 309–338 (2018; Zbl 1397.15008)], the PDF of the second basic example stated above is derived. The relevance of the Harish-Chandra integral on the unitary group is pointed out in order to compute the distribution of the diagonal entries of the random matrix \(U_{p} A U_{p}^{†},\) where \(U_{p}\) is the \(p\times n\) matrix defined by the first \(p\) rows of \(U\in \mathrm{U}(n)\), chosen with Haar measure.
Finally, some remarks concerning the random matrix sum \(UAU^{†}+ VBV^{†}\) when \(A,B\) are real antisymmetric and \(U\) is real orthogonal are pointed out.

MSC:

15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
60B20 Random matrices (probabilistic aspects)
15B57 Hermitian, skew-Hermitian, and related matrices
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