# On the structure of certain matrix classes

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This thesis considers two problems in matrix theory. The first problem Is concerned with the matrix equation AXB=X where A and B are n x n doubly stochastic and we seek n x n doubly stochastic matrix solutions X. Of course the n x n matrix J[loweredn] = (1/n) is always a solution. We note that the set of doubly stochastic matrix solutions of this equation forms a compact convex set. We thus describe all doubly stochastic matrix solutions of the equation by specifying the vertices of this set. Finally, we give an example and show some applications of this equation. The second problem is concerned with nearly reducible and nearly decomposable matrices. Relationships between nearly reducible matrices, nearly decomposable matrices and graph theory are shown. In particular, through graph theory, It Is shown that nearly reducible and nearly decomposable matrices have a simple structural form. Problems of constructing nearly reducible and nearly decomposable matrices from nearly reducible and. nearly decomposable matrices of lower dimension are also discussed. Finally, some applications and properties of nearly reducible and nearly decomposable matrices are included. The thesis Is concluded by discussing several problems which are motivated by the thesis but which remain unanswered.