Identifying the spectrum of the sum of two given Hermitian matrices with
fixed eigenvalues is the famous Horn's problem.In this note, we investigate a
variant of Horn's problem, i.e., we identify the probability density function
(abbr. pdf) of the diagonals of the sum of two random Hermitian matrices with
given spectra. We then use it to re-derive the pdf of the eigenvalues of the
sum of two random Hermitian matrices with given eigenvalues via
\emph{derivative principle}, a powerful tool used to get the exact probability
distribution by reducing to the corresponding distribution of diagonal
entries.We can recover Jean-Bernard Zuber's recent results on the pdf of the
eigenvalues of two random Hermitian matrices with given eigenvalues. Moreover,
as an illustration, we derive the analytical expressions of eigenvalues of the
sum of two random Hermitian matrices from \rG\rU\rE(n) or Wishart ensemble by
derivative principle, respectively.We also investigate the statistics of
exponential of random matrices and connect them with Golden-Thompson
inequality, and partly answer a question proposed by Forrester. Some potential
applications in quantum information theory, such as uniform average quantum
Jensen-Shannon divergence and average coherence of uniform mixture of two
orbits,are discussed.Comment: 24 pages, LaTeX; a new result, i.e., Theorem 3.7, is added and
several references are include