Difference between revisions of "Partitioned Matrix"

Latest revision as of 18:33, 16 November 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 (Using the concet of the Schur Complement).

Further Resources

References

Block Matrix @ Wikipedia

↑ R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009

[1] R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009

[1]

@@ Line 3: / Line 3: @@
 == 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>
-== Tensor Representation ==
+If a matrix is partitioned into four blocks, it can be inverted blockwise (Using the concet of the [[wikipedia:Schur complement | Schur Complement]]).
-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 ==
+* [[Aggregation Matrix]]
+== Further Resources ==
+* [https://www.openriskacademy.com/course/view.php?id=70 Crash Course on Input-Output Model Mathematics]
+* [https://www.openriskacademy.com/course/view.php?id=64 Introduction to Input-Output Models using Python]
 == References ==