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

In summary, the Kronecker product of an argument X and a 2x2 matrix increases the dimensions of each argument X individually. This means that if each argument X is a scalar value, it will now become a 2x2 matrix. When performing the Kronecker product, the elements in the second diagonal are multiplied by a 2x2 matrix Q, while the elements in the first column are multiplied by a 2x2 matrix G. This results in each value producing a 2x2 matrix. However, for elements in the original matrix that were 0, there may be an alignment issue when forming the new resultant matrix. To address this, it is necessary to also perform the Kronecker product with
  • #1
pipebomb
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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
 
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  • #2
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.
 

1. What is the Kronecker product?

The Kronecker product is a mathematical operation that takes two matrices as inputs and produces a larger matrix as an output. It is denoted by the symbol ⊗ and is also known as the tensor product.

2. How is the Kronecker product calculated?

The Kronecker product is calculated by multiplying each element of one matrix with the entire other matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. The element at the intersection of a row in the first matrix and a column in the second matrix will be the product of the corresponding elements in those rows and columns.

3. What happens when the Kronecker product is performed on only a few elements in a matrix?

When the Kronecker product is performed on only a few elements in a matrix, the resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. However, only the elements used in the calculation will be present in the resulting matrix, with all other elements being zero.

4. How do you align resulting elements when performing the Kronecker product on only a few elements in a matrix?

The resulting elements can be aligned by following the same rules as when performing the Kronecker product on the entire matrices. Each element in the resulting matrix will be in the same position as the element in the first matrix that was used in the calculation, with all other elements being zero.

5. Are there any special considerations when performing the Kronecker product on only a few elements in a matrix?

Yes, when performing the Kronecker product on only a few elements in a matrix, it is important to ensure that the resulting matrix has the desired dimensions. If the resulting matrix is larger than necessary, it can be trimmed down to the desired size by removing any extra rows or columns that contain only zeros.

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