Similarity Is Preserved by Inverse and Transpose

Matrix similarity ($A=PBP^{-1}$) survives both inverting and transposing: if $A\sim B$ then $A^{-1}\sim B^{-1}$ and $A^{t}\sim B^{t}$. Plus every square matrix is similar to its own transpose, $A\sim A^{t}$. Chaining these rules lets you transform a given similarity into many others.