The paper concerns the space \(\Omega^{r}(n,\mathbf{F}) \), \(\mathbf{F=}\mathbb{R}\) or \(\mathbb{C}\), of \(r\)-generalized double stochastic matrix, i.e. the \(n\times n\) matrix having each row and column sums equal to \(r.\) The space \(\Omega^{r}(n,\mathbf{F}) \) is studied (mainly for \(r=0,1\)) at use the theory of Lie groups and Lie algebras. Firstly, it is noticed that \(\mathbf{ {GL}}_{1}(n,\mathbf{F}) :={\text\textbf{GL}}(n,\mathbf{F}) \cap\Omega ^{1}(n,\mathbf{F}) \) is a Lie group the Lie algebra of which is \(\Omega^{0}(n,\mathbf{F}) ;\) in consequence, the analogous relations hold for orthogonal and unitary groups \(\mathbf{O}_{1}(n) :=\mathbf{O}(n) \cap\Omega^{1}(n,\mathbf{F}) ,\) \(\mathfrak{o}_{0}(n) :=\mathfrak{o}(n,\mathbb{R}) \cap\Omega^{0}(n,\mathbb{R}) \) and \(\mathbf{U}_{1}(n) :=\mathbf{U}(n) \cap\Omega ^{1}(n,\mathbb{C}) ,\) \(\mathfrak{u}_{0}(n) :=\mathfrak{u}(n) \cap\Omega^{0}(n,\mathbb{C}).\) Next, it is proved that for any matrix \(V\in\mathbf{ {GL}}(n,\mathbf{F}) \) for which the first row is equal to \(1/\sqrt{n}(1,1,\dots,1) \) and each of the last \(n-1\) row sums are zero, the mapping \(\Phi_{V}:\mathbf{M}(n-1,\mathbf{F}) \to \Omega^{0}(n,\mathbf{F}) ,\) \(X\mapsto V^{-1}\left(\begin{smallmatrix} 0 & 0\\ 0 & X \end{smallmatrix}\right) V,\) is an algebra isomorphism. In consequence, if additionally \(V\) is orthogonal (for \(\mathbf{F}=\mathbb{R}\)), \(\Phi_{V}\) is an isomorphism of the Lie algebras \(\mathfrak{o}(n-1) \) and \(\mathfrak{o}_{0}(n) \) (analogously for \(\mathbf{F}=\mathbb{C}\) and unitary algebras) and \(\Phi_{V}\) is the differential of the isomorphism of the Lie group \(\phi _{V}:\mathbf{GL}(n-1,\mathbf{F}) \to\mathbf{GL} _{1}(n,\mathbf{F}) \) defined by the same formula (analogous facts concern orthogonal and unitary cases).
The aim of the paper is a presentation of applications to the inverse eigenvalue problem for symmetric doubly stochastic matrices. Firstly, it is proved that the adjoint action \(\mathbf{O}_{1}(n) \) on its Lie algebra \(\mathfrak{o}_{0}(n) \) is equivalent to the adjoint action of \(\mathbf{O}(n-1) \) on \(\mathfrak{o}(n-1) \) (analogous for unitary case). It implies that for \(A\in\Delta_{n}^{s}\) (the set of nonnegative symmetric elements in \(\Omega^{1}(n,\mathbb{R}) \)) and the orbits \(\mathbf{O}(n) \cdot A\) and \(\mathbf{O}_{1}(n) \cdot A\) through \(A\) of the actions of \(\mathbf{O}(n) \) and \(\mathbf{O}_{1}(n) \) on \(\mathbf{M}_{s}(n,\mathbb{R}) ,\) the equality \(\mathbf{O}(n) \cdot A\cap\Delta_{n}^{s}=\mathbf{O}_{1}(n) \cdot A\cap\Delta_{n}^{s}\) holds. { }Lastly the author describes how the orbits of the action of \(\mathbf{O}_{1}(3) \) on \(\Omega_{s}^{1}(3,\mathbb{R}) \) intersect \(\Delta_{s}^{3}.\) Namely using the map \(E:\mathbf{M}_{s}(n,\mathbb{R}) \to\mathbb{R}^{n},\) \(E(X) =(\lambda_{1},\dots,\lambda_{n}) \) where \((\lambda_{1},\dots,\lambda_{n}) \) is the decreasingly ordered set of the eigenvalues of the matrix \(X\) and \(\mathfrak{\Theta}_{n}^{s}=E[ \Delta_{n}^{s}] \) the author obtains the equality \(\mathfrak{\Theta}_{3}^{s}=\text{*Conv}[(1,1,1) ,(1,1,-1) ,(1,-\frac{1}{2},-\frac{1}{2}) ]\).