In mathematics , especially in linear algebra and matrix theory , the duplication matrix and the elimination matrix are linear transformations used for transforming half-vectorizations of matrices into vectorizations or (respectively) vice versa.

Duplication matrix [ edit ]

The duplication matrix ${\displaystyle D_{n}}$ is the unique ${\displaystyle n^{2}\times {\frac {n(n+1)}{2}}}$ matrix which, for any ${\displaystyle n\times n}$ symmetric matrix ${\displaystyle A}$, transforms ${\displaystyle \mathrm {vech} (A)}$ into ${\displaystyle \mathrm {vec} (A)}$:

${\displaystyle D_{n}\mathrm {vech} (A)=\mathrm {vec} (A)}$.

For the ${\displaystyle 2\times 2}$ symmetric matrix ${\displaystyle A=\left[{\begin{smallmatrix}a&b\\b&d\end{smallmatrix}}\right]}$, this transformation reads

${\displaystyle D_{n}\mathrm {vech} (A)=\mathrm {vec} (A)\implies {\begin{bmatrix}1&0&0\\0&1&0\\0&1&0\\0&0&1\end{bmatrix}}{\begin{bmatrix}a\\b\\d\end{bmatrix}}={\begin{bmatrix}a\\b\\b\\d\end{bmatrix}}}$

The explicit formula for calculating the duplication matrix for a ${\displaystyle n\times n}$ matrix is:

${\displaystyle D_{n}^{T}=\sum \limits _{i\geq j}u_{ij}(\mathrm {vec} T_{ij})^{T}}$

Where:

• ${\displaystyle u_{ij}}$ is a unit vector of order ${\displaystyle {\frac {1}{2}}n(n+1)}$ having the value ${\displaystyle 1}$ in the position ${\displaystyle (j-1)n+i-{\frac {1}{2}}j(j-1)}$ and 0 elsewhere;
• ${\displaystyle T_{ij}}$ is a ${\displaystyle n\times n}$ matrix with 1 in position ${\displaystyle (i,j)}$ and ${\displaystyle (j,i)}$ and zero elsewhere

Here is a C++ function using Armadillo (C++ library) :

arma
::
mat
 duplication_matrix
(
const
 int
 &
n
)
 {

    arma
::
mat
 out
((
n
*
(
n
+
1
))
/
2
,
 n
*
n
,
 arma
::
fill
::
zeros
);

    for
 (
int
 j
 =
 0
;
 j
 <
 n
;
 ++
j
)
 {

        for
 (
int
 i
 =
 j
;
 i
 <
 n
;
 ++
i
)
 {

            arma
::
vec
 u
((
n
*
(
n
+
1
))
/
2
,
 arma
::
fill
::
zeros
);

            u
(
j
*
n
+
i
-
((
j
+
1
)
*
j
)
/
2
)
 =
 1.0
;

            arma
::
mat
 T
(
n
,
n
,
 arma
::
fill
::
zeros
);

            T
(
i
,
j
)
 =
 1.0
;

            T
(
j
,
i
)
 =
 1.0
;

            out
 +=
 u
 *
 arma
::
trans
(
arma
::
vectorise
(
T
));

        }

    }

    return
 out
.
t
();

}

Elimination matrix [ edit ]

An elimination matrix ${\displaystyle L_{n}}$ is a ${\displaystyle {\frac {n(n+1)}{2}}\times n^{2}}$ matrix which, for any ${\displaystyle n\times n}$ matrix ${\displaystyle A}$, transforms ${\displaystyle \mathrm {vec} (A)}$ into ${\displaystyle \mathrm {vech} (A)}$:

${\displaystyle L_{n}\mathrm {vec} (A)=\mathrm {vech} (A)}$.  ^[1]

By the explicit (constructive) definition given by Magnus & Neudecker (1980) , the ${\displaystyle {\frac {1}{2}}n(n+1)}$ by ${\displaystyle n^{2}}$ elimination matrix ${\displaystyle L_{n}}$ is given by

${\displaystyle L_{n}=\sum _{i\geq j}u_{ij}\mathrm {vec} (E_{ij})^{T}=\sum _{i\geq j}(u_{ij}\otimes e_{j}^{T}\otimes e_{i}^{T}),}$

where ${\displaystyle e_{i}}$ is a unit vector whose ${\displaystyle i}$-th element is one and zeros elsewhere, and ${\displaystyle E_{ij}=e_{i}e_{j}^{T}}$.

Here is a C++ function using Armadillo (C++ library) :

arma
::
mat
 elimination_matrix
(
const
 int
 &
n
)
 {

    arma
::
mat
 out
((
n
*
(
n
+
1
))
/
2
,
 n
*
n
,
 arma
::
fill
::
zeros
);

    for
 (
int
 j
 =
 0
;
 j
 <
 n
;
 ++
j
)
 {

        arma
::
rowvec
 e_j
(
n
,
 arma
::
fill
::
zeros
);

        e_j
(
j
)
 =
 1.0
;

        for
 (
int
 i
 =
 j
;
 i
 <
 n
;
 ++
i
)
 {

            arma
::
vec
 u
((
n
*
(
n
+
1
))
/
2
,
 arma
::
fill
::
zeros
);

            u
(
j
*
n
+
i
-
((
j
+
1
)
*
j
)
/
2
)
 =
 1.0
;

            arma
::
rowvec
 e_i
(
n
,
 arma
::
fill
::
zeros
);

            e_i
(
i
)
 =
 1.0
;

            out
 +=
 arma
::
kron
(
u
,
 arma
::
kron
(
e_j
,
 e_i
));

        }

    }

    return
 out
;

}

For the ${\displaystyle 2\times 2}$ matrix ${\displaystyle A=\left[{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right]}$, one choice for this transformation is given by

${\displaystyle L_{n}\mathrm {vec} (A)=\mathrm {vech} (A)\implies {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}a\\c\\b\\d\end{bmatrix}}={\begin{bmatrix}a\\c\\d\end{bmatrix}}}$.

Notes [ edit ]

References [ edit ]

• Magnus, Jan R.; Neudecker, Heinz (1980), "The elimination matrix: some lemmas and applications" , SIAM Journal on Algebraic and Discrete Methods , 1 (4): 422?449, doi : 10.1137/0601049 , ISSN   0196-5212 .
• Jan R. Magnus and Heinz Neudecker (1988), Matrix Differential Calculus with Applications in Statistics and Econometrics , Wiley. ISBN   0-471-98633-X .
• Jan R. Magnus (1988), Linear Structures , Oxford University Press. ISBN   0-19-520655-X