An Inverse Matrix Formula in the RightQuantum Algebra The rightquantum algebra was introduced recently by Garoufalidis, Le and Zeilberger in their quantum generalization of the MacMahon master theorem. A bijective proof of this identity due to Konvalinka and Pak, and also the recent proof of the rightquantum Sylvesteru0027s determinant identity, make heavy use of a bijection related to the first fundamental transformation on words introduced by Foata. This paper makes explicit the connection between this transformation and rightquantum linear algebra identities; we give a new, bijective proof of the rightquantum matrix inverse theorem, we show that similar techniques prove the rightquantum Jacobi ratio theorem, and we use the matrix inverse formula to find a generalization of the (rightquantum) MacMahon master theorem.

Selfconcordance and matrix monotonicity with applications to quantum entanglement problems Abstract   Let V be a Euclidean Jordan algebra and xcexa9 be a cone of invertible squares in V. Suppose that     g  :   R     xe2x86x92  R     is a matrix monotone function on the positive semiaxis which naturally induces a function      g  xcbx9c   :   xcexa9   xe2x86x92  V    . We show that     xe2x88x92   g  xcbx9c      is compatible (in the sense of Nesterovxe2x80x93Nemirovski) with the standard selfconcordant barrier     B  (  x  )    xe2x88x92  ln  det  (  x  )     on xcexa9. As a consequence, we show that for any cxc2xa0xe2x88x88xc2xa0xcexa9, the functions of the form     xe2x88x92  tr  (  c   g  xcbx9c    (  x  )   )    B   (  x  )      are selfconcordant on xcexa9. In particular, the function     x  xe2x86xa6  xe2x88x92  tr  (  c  ln  x  )     is a selfconcordant barrier function on xcexa9. Using these results, we apply a longstep pathfollowing algorithm developed in L.xc2xa0Faybusovich and C.xc2xa0Zhou Longstep pathfollowing algorithm for solving symmetric programming problems with nonlinear objective functions. Comput Optim Appl, 72(3):769795, 2019 to a number of important optimization problems arising in quantum information theory. Results of numerical experiments and comparisons with existing methods are presented.