Let $X$ be a real, finite-dimensional vector space and $A, B, C,$ and $D$ be matrices on $X$. I'm interested in the similarity classes of the block matrices $$ \begin{bmatrix} A & B\\ C & D\\ \end{bmatrix} $$ restricted to transformations of the form $$ \begin{bmatrix} A & B\\ C & D\\ \end{bmatrix} \mapsto \begin{bmatrix} I & 0\\ 0 & T\\ \end{bmatrix} \begin{bmatrix} A & B\\ C & D\\ \end{bmatrix} \begin{bmatrix} I & 0\\ 0 & T^{-1}\\ \end{bmatrix}, $$ where $T$ is an invertible transformation on $X$. Specifically, under what conditions, if any, does there exist a $T$ such that we have $$ \begin{bmatrix} I & 0\\ 0 & T\\ \end{bmatrix} \begin{bmatrix} A & B\\ C & D\\ \end{bmatrix} \begin{bmatrix} I & 0\\ 0 & T^{-1}\\ \end{bmatrix} = \begin{bmatrix} E & F\\ G & E+F-G\\ \end{bmatrix} $$ for some matrices $E,F,$ and $G$ on $X$. I would appreciate being pointed to any relevant literature.

  • $\begingroup$ Is there any reason why $B, C$ should be square? In other words, is there any reason why you can't have two spaces $X, Y$ with $A \in \text{End}(Y), B \in \mathcal{L}(X, Y), C \in \mathcal{L}(Y, X), D \in \text{End}(X)$? $\endgroup$ – user44191 Jun 10 '17 at 23:40
  • $\begingroup$ @user44191 This question comes from an application to Linear Quadratic Gaussian (LQG) optimal control. In this application, while it is theoretically possible to have two spaces, practical considerations eliminate such cases. Thus you may always assume $\begin{bmatrix} A & B \\ C& D\\ \end{bmatrix}$ is an endomorphism (specifically a real valued matrix) of $X \oplus X$. $\endgroup$ – JMJ Jun 10 '17 at 23:44
  • $\begingroup$ Even besides practical considerations, the desired bottom-right corner is $E+F-G$. It's not clear what the "right" analog of this would be if $F$ and $G$ weren't square, and of the same size as $E$. $\endgroup$ – Nathaniel Johnston Jun 11 '17 at 0:13
  • $\begingroup$ @NathanielJohnston True; I was focusing on the classification part first. The classification should be related to the representation theory of the 2-vertex quiver with one of each possible arrow (2 loops, 2 between the vertices); this representation theory is "wild" if I remember correctly - classification of indecomposables isn't nice. $\endgroup$ – user44191 Jun 11 '17 at 0:27

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