matricization

Matricization, also known as unfolding, is a process in multilinear algebra that converts a tensor into a matrix by rearranging its elements while preserving its mode structure. Given a tensor X of order N with dimensions I1 x I2 x ... x IN, the mode-n matricization X_(n) yields a matrix of size In x (I1 I2 ... I_{n-1} I_{n+1} ... IN). Each mode-n fiber (a vector obtained by fixing all indices except n) becomes a column of X_(n); equivalently, X_(n) arranges the slices perpendicular to mode n into columns.

Common convention defines the columns of X_(n) according to a chosen lexicographic ordering of the indices other than n. There are several conventions in use, and different software libraries may adopt different ones, but the essential idea is to represent the tensor with respect to a given mode as a two-dimensional matrix to enable matrix-based computations.

For a concrete example, a third-order tensor X in R^{I x J x K} has three matricizations: X_(1) is I x JK, X_(2) is J x IK, and X_(3) is K x IJ. Each one unfolds the tensor along the corresponding mode, turning mode-n fibers into the columns of the resulting matrix.

Matricization is a fundamental tool in tensor decompositions. In Tucker and CANDECOMP/PARAFAC (CP) decompositions, the factor matrices are often obtained by applying singular value decompositions to the mode-n unfoldings. The higher-order singular value decomposition (HOSVD) also relies on X_(n) to compute canonical factor matrices. The operation is not unique due to varying conventions, but it provides a bridge from tensor analysis to standard matrix techniques.