Algebraic spectral problems

Preorders on canonical families of modules of finite length

“…REMARK 2.7. There are many other proofs of Kronecker's theorem on canonical pairs of matrices under equivalence, for example [3], [5], [6], [10], [11], [14], [16], [17], and [18]. Some applications of the theorem can be found in [1], [2], [10], and [12].…”

“…So by (c), C/. contains a submodule Y (say) of type II 3 e J[ and II, does not precede II 3 . Since Y is a submodule of V, II 3 e S. This contradicts the choice of II,.…”

“…The space W is called the range or target space, while V is the domain space of ( F , W) of ( F , W). Kronecker modules can be considered as modules over a finite-dimensional AT-algebra, see, for example, [7, (3)) is an isomorphism of V onto W. The map from K 2 x V to V that takes (e,v) to ^"'(eow) is bilinear and so makes {V, V) a module. Let id be the identity map on V. Then (id, T~l) is an isomorphism from (V, W) to {V, V).…”

“…(For n > 3 , see [8], and for n = 1, see [13].) We assume that the matrices have entries in an algebraically closed field, K. Following [3], we replace the pairs of matrices by pairs of linear transformations from an ^-dimensional vector space V to an r-dimensional vector space W. By taking linear combinations, we get from each such pair of linear transformations a A"-bilinear map, o, from K 2 x V to W. By linearity, it is enough to specify the map on a basis (a, b) of K 2 and on a basis of V. The pair ( F , W) together with the bilinear map is a Kronecker module. The space W is called the range or target space, while V is the domain space of ( F , W) of ( F , W).…”

A bound on the rank of purely simple systems

“…Aronszajn became interested in these representations because of his investigation of finite-dimensional perturbations of spectral problems. He and Fixman in [1] call a representation of (1) a system. They prove results analogous to those in the theory of abelian groups.…”