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Abstract
The matrix bidiagonal form is analysed and some applications to linear system theory are shown. In the first part, it is shown that the reduction of a given matrix A to the MBF is immediate in the very common situation in which a vector b such that (A, b) is controllable is known. In the second part, some applications are presented such as the evaluation of a function of a matrix and the solution of the Lyapunov equation.