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

[pipebomb]

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