A generalized doubly stochastic matrix is an \(n\times n\) matrix such that each row and column sum is equal to a given \(x\neq 0\) in a field F, where char (F)\(\nmid n\). \(J_ n\) is the \(n\times n\) matrix such that if e is the unit in F and if n is also used to denote the n-fold sum \(e+e+...+e\), then every entry in \(J_ n\) is \(n^{-1}\). Operations \(\oplus\), \(\circ\), and \(\otimes\) defined on the generalized doubly stochastic matrices by \(A\oplus B=A+B-xJ_ n\), \(\alpha \circ A=\alpha A+(1-\alpha)xJ_ n\), and \(A\otimes B=AB+x(1-x)J_ n\), respectively, make the set of generalized doubly stochastic matrices a total algebra of linear operators. A basis consisting of \((n-1)^ 2\) particular permutation matrices is given for the generalized doubly stochastic matrices. The work is an extension of some of the results of E. C. Johnsen and the reviewer.