Ranks of linear matrix pencils separate simultaneous similarity orbits

This paper solves the two-sided version and provides a counterexample to general of 2003 conjecture by Hadwin Larson. Consider evaluations linear matrix pencils $L=T_0+x_1T_1+\cdots+x_mT_m$ on tuples as $L(X_1,\dots,X_m)=I\otimes T_0+X_1\otimes T_1+\cdots+X_m\otimes T_m$. It is shown that ranks constitute collection separating invariants for simultaneous similarity tuples. That is, $m$-tuples $A$ $B$ $n\times n$ matrices are simultaneously similar if only $L(A)$ $L(B)$ equal all $L$ size $mn$. Variants this property also established symplectic, orthogonal, unitary similarity, left-right action groups. Furthermore, polynomial time algorithm orbit equivalence under special groups deduced.