Ranks of linear matrix pencils separate simultaneous similarity orbits

Abstract

This paper solves the two-sided version and provides a counterexample to the general version of the 2003 conjecture by Hadwin and Larson. Consider evaluations of linear matrix pencils $L=T_0+x_1{T}_1+\cdots +x_m{T}_m$ on matrix 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 of linear matrix pencils constitute a collection of separating invariants for simultaneous similarity of matrix tuples. That is, $m$-tuples $A$ and $B$ of $n\times n$ matrices are simultaneously similar if and only if the ranks of $L(A)$ and $L(B)$ are equal for all linear matrix pencils $L$ of size $mn$. Variants of this property are also established for symplectic, orthogonal, unitary similarity, and for the left-right action of general linear groups. Furthermore, a polynomial time algorithm for orbit equivalence of matrix tuples under the left-right action of special linear groups is deduced.

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• Towards a theory of non-commutative optimization: geodesic first and second order methods for moment maps and polytopes
• Invariant theory and scaling algorithms for maximum likelihood estimation