Elimination Matrix
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
and
matrix theory In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begi ...
, the duplication matrix and the elimination matrix are
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s used for transforming half-vectorizations of
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into vectorizations or (respectively) vice versa.


Duplication matrix

The duplication matrix D_n is the unique n^2 \times \frac matrix which, for any n \times n
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...
A , transforms \mathrm(A) into \mathrm(A): : D_n \mathrm(A) = \mathrm(A). For the 2 \times 2 symmetric matrix A=\left begin a & b \\ b & d \end\right/math>, this transformation reads : D_n \mathrm(A) = \mathrm(A) \implies \begin 1&0&0 \\ 0&1&0 \\ 0&1&0 \\ 0&0&1 \end \begin a \\ b \\ d \end = \begin a \\ b \\ b \\ d \end The explicit formula for calculating the duplication matrix for a n \times n matrix is: D^T_n = \sum \limits_ u_ (\mathrmT_)^T Where: * u_ is a unit vector of order \frac n (n+1) having the value 1 in the position (j-1)n+i - \fracj(j-1) and 0 elsewhere; * T_ is a n \times n matrix with 1 in position (i,j) and (j,i) and zero elsewhere Here is a
C++ C++ (pronounced "C plus plus") is a high-level general-purpose programming language created by Danish computer scientist Bjarne Stroustrup as an extension of the C programming language, or "C with Classes". The language has expanded significan ...
function using
Armadillo (C++ library) Armadillos (meaning "little armored ones" in Spanish) are New World placental mammals in the order Cingulata. The Chlamyphoridae and Dasypodidae are the only surviving families in the order, which is part of the superorder Xenarthra, along wi ...
: arma::mat duplication_matrix(const int &n)


Elimination matrix

An elimination matrix L_n is a \frac \times n^2 matrix which, for any n \times n matrix A, transforms \mathrm(A) into \mathrm(A): :L_n \mathrm(A) = \mathrm(A), Definition 3.1 By the explicit (constructive) definition given by , the \fracn(n+1) by n^2 elimination matrix L_n is given by :L_n = \sum_ u_ \mathrm(E_)^T = \sum_ (u_\otimes e_j^T \otimes e_i^T), where e_i is a unit vector whose i-th element is one and zeros elsewhere, and E_ = e_ie_j^T. Here is a
C++ C++ (pronounced "C plus plus") is a high-level general-purpose programming language created by Danish computer scientist Bjarne Stroustrup as an extension of the C programming language, or "C with Classes". The language has expanded significan ...
function using
Armadillo (C++ library) Armadillos (meaning "little armored ones" in Spanish) are New World placental mammals in the order Cingulata. The Chlamyphoridae and Dasypodidae are the only surviving families in the order, which is part of the superorder Xenarthra, along wi ...
: arma::mat elimination_matrix(const int &n) For the 2 \times 2 matrix A = \left begin a & b \\ c & d \end\right/math>, one choice for this transformation is given by : L_n \mathrm(A) = \mathrm(A) \implies \begin 1&0&0&0 \\ 0&1&0&0 \\ 0&0&0&1 \end \begin a \\ c \\ b \\ d \end = \begin a \\ c \\ d \end.


Notes


References

*. *Jan R. Magnus and Heinz Neudecker (1988), ''Matrix Differential Calculus with Applications in Statistics and Econometrics'', Wiley. . *Jan R. Magnus (1988), ''Linear Structures'', Oxford University Press. {{ISBN, 0-19-520655-X Matrices de:Eliminationsmatrix