In multilinear algebra, mode-m flattening[1][2][3], also known as matrixizing, matricizing, or unfolding,[4] is an operation that reshapes a multi-way array into a matrix denoted by (a two-way array).
Matrixizing may be regarded as a generalization of the mathematical concept of vectorizing.
Definition
The mode-m matrixizing of tensor is defined as the matrix . As the parenthetical ordering indicates, the mode-m column vectors are arranged by sweeping all the other mode indices through their ranges, with smaller mode indexes varying more rapidly than larger ones; thus[1]
where and
By comparison, the matrix that results from an unfolding[4] has columns that are the result of sweeping through all the modes in a circular manner beginning with mode m + 1 as seen in the parenthetical ordering. This is an inefficient way to matrixize.[citation needed]