Generalized matrix functions, determinant and permanent

Permanent Uncertainty: on the Quantum Evaluation of the Determinant and the Permanent of a Matrix

We investigate the possibility of evaluating permanents and determinants of matrices by quantum computation. All current algorithms for the evaluation of the permanent of a real matrix have exponential time complexity and are known to be in the class P #P. Any method to evaluate or approximate the permanent is thus of fundamental interest to complexity theory. Permanents and determinants of mat...

Computation of Generalized Matrix Functions

We develop numerical algorithms for the efficient evaluation of quantities associated with generalized matrix functions [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra, 1(2), 1973, pp. 163–171]. Our algorithms are based on Gaussian quadrature and Golub–Kahan bidiagonalization. Block variants are also investigated. Numerical experiments are performed to illustrate the effectiven...

Determinant versus permanent

Determinant Versus Permanent

We study the problem of expressing permanents of matrices as determinants of (possibly larger) matrices. This problem has close connections to the complexity of arithmetic computations: the complexities of computing permanent and determinant roughly correspond to arithmetic versions of the classes NP and P respectively. We survey known results about their relative complexity and describe two re...

determinant of the hankel matrix with binomial entries

abstract in this thesis at first we comput the determinant of hankel matrix with enteries a_k (x)=?_(m=0)^k??((2k+2-m)¦(k-m)) x^m ? by using a new operator, ? and by writing and solving differential equation of order two at points x=2 and x=-2 . also we show that this determinant under k-binomial transformation is invariant.

Permanent versus determinant, obstructions, and Kronecker coefficients

We give an introduction to some of the recent ideas that go under the name “geometric complexity theory”. We first sketch the proof of the known upper and lower bounds for the determinantal complexity of the permanent. We then introduce the concept of a representation theoretic obstruction, which has close links to algebraic combinatorics, and we explain some of the insights gained so far. In p...