Difference between revisions of "Partitioned Matrix"
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| + | == 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. | ||
| + | |||
| + | == References == | ||
| + | * [[wikipedia:Block matrix | Block Matrix @ Wikipedia]] | ||
| + | <references/> | ||
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| + | [[Category:EEIO]] | ||
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| + | {{#set:Has Formula = HAS_FORMULA}} | ||
| + | |||
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Revision as of 16:32, 28 February 2022
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.
