Is there any transformation between "nonregular" polynomial matrices that preserves except of the finite and infinite zero structure, their left and right minimal indices as well? (Except the case of matrix pencils where the strict equivalence is the answer)
Transformation linear or nonlinear of a matrix will not preserve the structure of the matrix. Even for linear ( similarity) transformation, only the eigen values are preserved in addition to the structural parameters such as dimensions only are preserved. In fact, most of the cases, the transformation is to modify the structure of the matrix that may make many properties apparent by inspection. For example diagonalisation yields eigen values on the diagonal, ease of inversion, multiplication etc.
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