Relations between the Permutations and the Matrix Norm in Denumerable Infinite Vector Folding to Semi-denumerable Infinite Matrices

This work focuses on the folding and rank issues for the denumerable infinite vector foldings to denumerable semi infinite matrices. The vector to be folded is assumed to have denumerably infinite number of elements while the produced matrix is assumed to be composed of a finite number of rows and denumerably infinite number of columns as we have done in some other works of us. The vector folding operation locates the elements of the given vector to the available positions of the target matrix. However, this action is not unique and different patterns for the element locating procedure can be used to get different resulting matrices whose ranks may differ from case to case. This work involves certain discussions about the pattern definitions via element permutations and their effects on the resulting matrix rank.