Difference between revisions of "Partitioned Matrix"
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== Usage == | == 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) | + | 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.<ref>R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009</ref> |
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| + | If a matrix is partitioned into four blocks, it can be inverted blockwise (Schur Complement) | ||
== Tensor Representation == | == Tensor Representation == | ||
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== See Also == | == See Also == | ||
* [[Aggregation Matrix]] | * [[Aggregation Matrix]] | ||
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| + | == Further Resources == | ||
| + | * [https://www.openriskacademy.com/mod/page/view.php?id=800 Crash Course on Input-Output Model Mathematics] | ||
== References == | == References == | ||
Revision as of 13:50, 18 September 2023
Definition
A Partitioned Matrix (or Block Matrix) is the general mathematical structure used prominently in the context of Multiregional 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.[1]
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.
See Also
Further Resources
References
- ↑ R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009
