@inbook{10.2307/j.ctt6wpz0p.17, ISBN = {9780691161853}, URL = {http://www.jstor.org/stable/j.ctt6wpz0p.17}, abstract = {Here, we present applications to polynomial matrix equations, algebraic Riccati equations, and linear quadratic regulators. Without attempting to develop indepth exposition of the topics (this would take us too far afield), we present these applications in basic forms. Maximal invariant semidefinite or neutral subspaces will play a key role.The approach to studying polynomial equations using companion matrices of Section 5.12 extends to polynomial matrix equations. Consider the matrix equation${Z^n} + {A_{n - 1}}{Z^{n - 1}} - \cdots + {A_1}Z + {A_0} - 0$, (14.1.1)where${A_0}, \ldots ,{A_{n - 1}} \in {H^{m \times m}}$are given and$Z \in {H^{m \times m}}$is the unknown matrix. LetCbe theblock companion matrixcorresponding to equation (14.1.1):$\left( {\begin{array}{*{20}{c}} 0 & {{I_m}} & 0 & 0 & \cdots & 0 \\ 0 & 0 & {{I_m}} & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \cdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & {{I_m}} \\ { - {A_0}} & { - {A_1}} & { - {A_2}} & { - {A_3}} & \cdots & { - {A_{n - 1}}} \\ \end{array}} \right) \in {H^{(mn) \times (mn)}}$Theorem 14.1.1. There exists a one-to-one}, bookauthor = {Leiba Rodman}, booktitle = {Topics in Quaternion Linear Algebra}, pages = {328--338}, publisher = {Princeton University Press}, title = {Matrix equations}, year = {2014} }