> [!NOTE] Definition (Elementary Matrix) >Let $A,B$ be [[Real Matrices|real matrices]] of order $m\times n.$ Then $A$ and $B$ are equivalent if and only if there exists real square matrices $P\in\mathbb{R}^{mm}$ and $Q\in\mathbb{R}^{nn}$ so that $B=P^{-1}AQ.$ # Properties By [[Equivalence of Real Matrices is Equivalence Relation]], ...