Duplication matrix

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, especially in linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

and matrix theory, the duplication matrix and the elimination matrix are linear transformation

Linear transformation

In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

s used for transforming half-vectorizations of matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

into vectorizations

Vectorization (mathematics)

In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector...

or (respectively) vice-versa.

The duplication matrix D_{n is the unique n² × n(n+1)/2 matrix which, for any n × n symmetric matrix A, transforms vech(A) into vec(A):

D_n vech(A) = vec(A).

For the 2×2 symmetric matrix A = , this transformation reads

Elimination matrix
The elimination matrix L_{n is the unique n(n+1)/2 × n² matrix which, for any n × n matrix A, transforms vec(A) into vech(A):

L_n vec(A) = vech(A).

For the 2×2 matrix A = , this transformation reads

.

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