Essentially doubly stochastic matrices

In this paper we extend the general theory of essentially doubly stochastic (e.d.s.) matrices begun in earlier papers in this series. We complete the investigation in one direction by characterizing all of the algebra isomorphisms between the algebra of e.d.s. matrices of order n over a field F,E_n(F), and the total algebra of matrices of order n - 1over F,M_n-1(F) We then develop some of the theory when Fis a field with an involution. We show that for any e,f§Fof norm 1,e≠f every e.d.s. matrix in E_n(F) is a unique e.d.s. sum of an e.d.s. e-hermitian matrix and an e.d.s. f-hermitian matrix in E_n(F) Next, we completely determine the cases for which there exists an above-mentioned matrix algebra isomorphism preserving adjoints. Finally, we consider cogredience in E_n(F) and show that when such an adjoint-preserving isomorphism exists and char M_n(F) two e.d.s. e-hermitian matrices which are cogredient in M_n(F) are also cogredient in E_n(F). Using this result, we obtain simple canonical forms for cogredience of e.d.s. e-hermitian matrices in E_n(F) when Fsatisfies special conditions. This ncludes the e.d.s. skew-symmetric matrices, where the involution is trivial and E = -1.