The isomorphisms between the upper and lower triangular matrix algebras
Since matrix equations with triangular matrices are easier to solve, the triangular matrices are very important in mathematics. For instance, the LU decomposition gives an algorithm to decompose any invertible matrix A into two triangular factors: a lower triangle matrix L and an upper triangle matrix U. Moreover, the inverse of a triangular matrix is also triangular. Also the product of two lower triangular matrices produces a lower triangular matrix, the same apply for upper triangular matrices. A lot of important notions, such as the determinant, the eigenvalue problem and many others are easy to handle when we are working with triangular matrices. Each of the two classes—the class of the lower triangular matrices and the class of upper triangular matrices—form an algebra. This thesis studies an important relation between the algebras of the upper and lower triangular matrices, the isomorphism. As we know, isomorphisms are used in representation theorems, where an abstract structure is similar to a concrete structure. In our case we are going to find “THE” algebras isomorphisms between two algebras in question. The preliminaries provides basic definitions, theorems, and some incentives involved in adopting this topic. The following chapters will be dedicated for finding the isomorphism, the method used in solving the problem is a comparison of the dimensions of the right and left annihilators of some specific matrices.^
Fawaz, Zahi, "The isomorphisms between the upper and lower triangular matrix algebras" (2011). ETD Collection for University of Texas, El Paso. AAI1494346.