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.(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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