Abstract. We show that several classes of matrix equations can be reduced to solving a quadratic matrix equation of the kind $AX^2+BX+C=0$ where $A,B,C,X$ are $m\times m$ matrices or semi-infinite matrices. The problem of computing the minimal solution, if it exists, of the latter equation is reduced to computing the matrix coefficients $H_0$ and $H_1$ of the Laurent matrix series $\sum_{i=-\infty}^{+\infty} z^iH_i$ such that $H(z)(zA+B+z^{-1}C)=I$. Known algorithms for this computation are revisited in terms of operations among block Toeplitz matrices and new algorithms are introduced. An application to the solution of Non-Skip-Free Markov chains is shown.

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