Inversion of the Multiplicative Matrix Compound Operator
This solves a theoretical inverse problem for matrix compounds, but is purely mathematical with no immediate application or empirical validation.
The paper characterizes the set of matrices whose kth multiplicative compound equals a given matrix M, showing that if rank(M) ≤ 1 there are infinitely many such matrices, and if rank(M) > 1 there is a unique matrix up to sign. An algorithm with analyzed time complexity is provided.
We study the problem of determining a matrix whose $k$th multiplicative compound is a prescribed matrix~$M$. The cardinality of the set of matrices whose $k$th multiplicative compound equals~$M$ 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~$M$. An algorithm for finding a matrix whose compound equals~$M$ is detailed, and its time complexity is analyzed.