Using elementary techniques, an algorithmic procedure to construct skew-symmetric matrices realizing the real irreducible representations of so(3) is developed. We further give a simple criterion that enables one to deduce the decomposition of an arbitrary real representation R of so(3) into real irreducible components from the characteristic polynomial of an arbitrary representation matrix.

We describe the MRS solver, a Minimal Residual method based on the Lanczos algorithm that solves problems from the important class of linear systems with a shifted skew-symmetric coefficient matrix using short vector recurrences. The MRS solver is theoretically compared with other Krylov solvers and illustrated by some numerical experiments.

Two algorithms are proposed for computing the maximum degree of a principal minor of specified order of a skew-symmetric rational function matrix. The algorithms are developed in the framework of valuated ∆matroid of Dress and Wenzel, and are valid also for valuated ∆-matroids in general.

Using an appropriate notion of equivalence, those classical Hamiltonian systems which admit a first integral of motion polynomial of degree one in momentum are classified. The classification is effected by means of finding a normal form for a skew-symmetric matrix under the action of orthogonal symmetry.

The map which takes a square matrix to its tropical eigenvalue-eigenvector pair is piecewise linear. We determine the cones of linearity of this map. They are simplicial but they do not form a fan. Motivated by statistical ranking, we also study the restriction of that cone decomposition to the subspace of skew-symmetric matrices.

The ideals generated by pfaffians of mixed size contained in a subladder of a skew-symmetric matrix of indeterminates define arithmetically Cohen-Macaulay, projectively normal, reduced and irreducible projective varieties. We show that these varieties belong to the G-biliaison class of a complete intersection. In particular, they are glicci.

For inner products de ned by a symmetric inde nite matrix p;q, canonical forms for real or complex p;q-Hermitian matrices, p;q-skew Hermitian matrices and p;q-unitary matrices are studied under equivalence transformations which keep the class invariant.

In 8] and 9] W. Mackens and the present author presented two generalizations of a method of Cybenko and Van Loan 4] for computing the smallest eigenvalue of a symmetric, positive deenite Toeplitz matrix. Taking advantage of the symmetry or skew symmetry of the corresponding eigenvector both methods are improved considerably.