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Kronecker product on only a few elements in a matrix: How to align resulting elements

  1. Jun 14, 2010 #1
    The Kronecker product of an argument X and a 2x2 matrix, increases the dimensions of each argument X individually. If each argument X is a scalar value, it now becomes a 2x2 matrix.

    How are these arguments now aligned with each other and the other elements in the resultant matrix?

    For example:
    In a matrix T, if we have several values and a certain number of 0s like:
    T=
    [0 1 0 ]
    [0.3 0 0.7]
    [0.1 0 0 ]

    For the new matrix TT, we perform the Kronecker product of the values in the second diagonal (i.e. of 1 and of 0.7) with a 2x2 matrix Q which is

    Q is
    [0.8 0.2]
    [0.4 0.6]

    and the Kronecker product of the values in the first column (i.e. of 0.3 and of 0.1) with a 2x2 matrix G which is

    G is
    [1 1]
    [1 1]

    Each value i.e. 1, 0.7, 0.3, 0.1 now results into a 2x2 matrix

    How are the resultant Kronecker product values aligned into the new resultant matrix TT, since the 0s remain as single scalar vaules?

    With the dimensions of only a few elements increasing, there will be an alignment issue with regard to the existing unchenanged elements i.e. the 0 values. How do I align them correctly to form a new resultant matrix without losing context?

    Do I (by default) HAVE to perform Kronecker product with the 0s also? If so, which 2x2 matrix do I use for that (Q or G)?

    For further reference:

    This problem arises from trying to validate and repeat the mathematical model given in Equations (19) and (20) to form a new matrix given in Equation (18) in the publication:
    T. Issariyakul and E. Hossain, "Performance modeling and analysis of a class of ARQ protocols in multi-hop wireless networks, " IEEE Transactions on Wireless Communications, vol. 5, no. 12, Dec. 2006, pp. 3460-3468.

    Thanks
     
  2. jcsd
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