Inversion of the Multiplicative Matrix Compound Operator
arXiv:2605.27682
The paper investigates how to recover a matrix whose k‑th multiplicative compound equals a given matrix M, showing infinitely many solutions when rank(M) ≤ 1 and a unique solution up to sign when rank(M) > 1, and provides an algorithm with complexity analysis.
Abstract
We study the problem of determining a matrix whose th multiplicative compound, with , is a prescribed matrix . The cardinality of the set of matrices whose th multiplicative compound equals is characterized in terms of $\rank(M)$. On the one hand, if $\rank(M)\le 1$, it is shown that there exist infinitely many such matrices for which a complete characterization is determined. On the other hand, if $\rank(M)>1$, then there exists a unique matrix -- up to an overall sign -- whose compound is . An algorithm for finding a matrix whose compound equals is detailed, and its time complexity is analyzed.