Partitioned Matrix
Definition
A Partitioned Matrix (or Block Matrix) is the general mathematical structure used prominently in the context of Multiregion Input-Output Model. It divides n industries in the Input‐Output Model into subgroups.
Usage
A partioned matrix may group specific sectors within an economy or represent a sector-country decomposition in a Multiregional Input-Output Model. An MRIO model extends the standard IO matrix to a larger system where each industry in each country has a separate row and column. If a matrix is partitioned into four blocks, it can be inverted blockwise (Schur Complement)
Tensor Representation
An N th-order tensor is an element of the tensor product of N vector spaces, each of which has its own coordinate system. Slices are two-dimensional sections of a tensor, defined by fixing all but two indices. Matricization, also known as unfolding or flattening, is the process of reordering the elements of an N-th order array into a matrix. For instance, a 2 × 2 × 3 × 3 tensor can be arranged as a 6 × 6 matrix or a 2 × 18 matrix, and so on. It is also possible to vectorize a tensor; for example, 2 × 2 × 3 × 3 tensor can be arranged as a 36 dimensional vector.
