We prove that the families matrix powering, iterated matrix product, and adjoint matrix are VQP–complete, where VQP denotes Valiant’s class of quasipolynomial–computable families of multivariate polynomials. This proves a conjecture by Bürgisser [3, Conjecture 8.1].

We prove that the families matrix powering, iterated matrix product, and adjoint matrix are VQP{complete, where VQP denotes Valiant's class of quasipolynomial{computable families of multivariate polynomials. This proves a conjecture by B urgisser 3, Conjecture 8.1].

De nition. Let (V, 〈 , 〉) be a n-dimensional euclidean vector space and T : V −→ V a linear operator. We will call the adjoint of T , the linear operator S : V −→ V such that: 〈T (u), v〉 = 〈u, S(v)〉 , for all u, v ∈ V . Proposition 1. Let (V, 〈 , 〉) be a n-dimensional euclidean vector space and T : V −→ V a linear operator. The adjoint of T exists and is unique. Moreover, if E denotes an orthon...

An observation sensitivity method to identify targeted observations is implemented in the context of four dimensional variational (4D-Var) data assimilation. This methodology is compared with the well-established adjoint sensitivity method using a nonlinear Burgers equation as a test model. Automatic differentiation software is used to implement the first order adjoint model to calculate the gr...

An upper bound on operator norms of the adjoint matrix is presented, and special cases of the l 1 , l 2 and l 1 norms are investigated. The results are then used to obtain lower bounds on the smallest absolute value of an eigenvalue of a nonsingular matrix.

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A,B in the self-adjoint Jacobi operator H = AS + AS + B (with S the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E−, E+], E− < E+, we prove that A and B are certa...

In this paper a new family of companion forms associated to a regular polynomial matrix is presented. Similar results have been presented in a recent paper by M. Fiedler, where the scalar case is considered. It is shown that the new family of companion forms preserves both the finite and infinite elementary divisors structure of the original polynomial matrix, thus all its members can be seen a...

We review a class of matrix models whose degrees of freedom are matrices with anticommuting elements. We discuss the properties of the adjoint fermion one-, twoand gauge invariant D-dimensional matrix models at large-N and compare them with their bosonic counterparts which are the more familiar Hermitian matrix models. We derive and solve the complete sets of loop equations for the correlators ...

We calculate a two-parameter R-matrix which specialises to the trigonometric R-matrix of the minimal affinisation of the adjoint representation of each of the classical simple Lie algebras so(n) and sp(n).