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

  • Thread starter pipebomb
  • Start date
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
 

fresh_42

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It makes more sense to operate with tensor products, which the Kronecker product is, only written differently. Look at Wikipedia for the definition. It's hard to follow the above writings. You basically produce scalar multiples of a given matrix and arrange the copies again as a matrix.
 

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