duplication and elimination matrices

    Duplication and elimination matrices GudangMovies21 Rebahinxxi LK21

    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


    The duplication matrix




    D

    n




    {\displaystyle D_{n}}

    is the unique




    n

    2


    ×



    n
    (
    n
    +
    1
    )

    2




    {\displaystyle n^{2}\times {\frac {n(n+1)}{2}}}

    matrix which, for any



    n
    ×
    n


    {\displaystyle n\times n}

    symmetric matrix



    A


    {\displaystyle A}

    , transforms




    v
    e
    c
    h

    (
    A
    )


    {\displaystyle \mathrm {vech} (A)}

    into




    v
    e
    c

    (
    A
    )


    {\displaystyle \mathrm {vec} (A)}

    :





    D

    n



    v
    e
    c
    h

    (
    A
    )
    =

    v
    e
    c

    (
    A
    )


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

    .
    For the



    2
    ×
    2


    {\displaystyle 2\times 2}

    symmetric matrix



    A
    =

    [





    a


    b




    b


    d





    ]



    {\displaystyle A=\left[{\begin{smallmatrix}a&b\\b&d\end{smallmatrix}}\right]}

    , this transformation reads





    D

    n



    v
    e
    c
    h

    (
    A
    )
    =

    v
    e
    c

    (
    A
    )

    ⟹



    [



    1


    0


    0




    0


    1


    0




    0


    1


    0




    0


    0


    1



    ]




    [



    a




    b




    d



    ]


    =


    [



    a




    b




    b




    d



    ]




    {\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



    n
    ×
    n


    {\displaystyle n\times n}

    matrix is:





    D

    n


    T


    =

    ∑

    i
    ≥
    j



    u

    i
    j


    (

    v
    e
    c


    T

    i
    j



    )

    T




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


    Where:





    u

    i
    j




    {\displaystyle u_{ij}}

    is a unit vector of order





    1
    2


    n
    (
    n
    +
    1
    )


    {\displaystyle {\frac {1}{2}}n(n+1)}

    having the value



    1


    {\displaystyle 1}

    in the position



    (
    j
    −
    1
    )
    n
    +
    i
    −


    1
    2


    j
    (
    j
    −
    1
    )


    {\displaystyle (j-1)n+i-{\frac {1}{2}}j(j-1)}

    and 0 elsewhere;





    T

    i
    j




    {\displaystyle T_{ij}}

    is a



    n
    ×
    n


    {\displaystyle n\times n}

    matrix with 1 in position



    (
    i
    ,
    j
    )


    {\displaystyle (i,j)}

    and



    (
    j
    ,
    i
    )


    {\displaystyle (j,i)}

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


    Elimination matrix


    An elimination matrix




    L

    n




    {\displaystyle L_{n}}

    is a






    n
    (
    n
    +
    1
    )

    2


    ×

    n

    2




    {\displaystyle {\frac {n(n+1)}{2}}\times n^{2}}

    matrix which, for any



    n
    ×
    n


    {\displaystyle n\times n}

    matrix



    A


    {\displaystyle A}

    , transforms




    v
    e
    c

    (
    A
    )


    {\displaystyle \mathrm {vec} (A)}

    into




    v
    e
    c
    h

    (
    A
    )


    {\displaystyle \mathrm {vech} (A)}

    :





    L

    n



    v
    e
    c

    (
    A
    )
    =

    v
    e
    c
    h

    (
    A
    )


    {\displaystyle L_{n}\mathrm {vec} (A)=\mathrm {vech} (A)}

    .
    By the explicit (constructive) definition given by Magnus & Neudecker (1980), the





    1
    2


    n
    (
    n
    +
    1
    )


    {\displaystyle {\frac {1}{2}}n(n+1)}

    by




    n

    2




    {\displaystyle n^{2}}

    elimination matrix




    L

    n




    {\displaystyle L_{n}}

    is given by





    L

    n


    =

    ∑

    i
    ≥
    j



    u

    i
    j



    v
    e
    c

    (

    E

    i
    j



    )

    T


    =

    ∑

    i
    ≥
    j


    (

    u

    i
    j


    ⊗

    e

    j


    T


    ⊗

    e

    i


    T


    )
    ,


    {\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




    e

    i




    {\displaystyle e_{i}}

    is a unit vector whose



    i


    {\displaystyle i}

    -th element is one and zeros elsewhere, and




    E

    i
    j


    =

    e

    i



    e

    j


    T




    {\displaystyle E_{ij}=e_{i}e_{j}^{T}}

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

    For the



    2
    ×
    2


    {\displaystyle 2\times 2}

    matrix



    A
    =

    [





    a


    b




    c


    d





    ]



    {\displaystyle A=\left[{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right]}

    , one choice for this transformation is given by





    L

    n



    v
    e
    c

    (
    A
    )
    =

    v
    e
    c
    h

    (
    A
    )

    ⟹



    [



    1


    0


    0


    0




    0


    1


    0


    0




    0


    0


    0


    1



    ]




    [



    a




    c




    b




    d



    ]


    =


    [



    a




    c




    d



    ]




    {\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




    References


    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

Kata Kunci Pencarian: duplication and elimination matrices

duplication and elimination matrices