Multiplication of two matrices ? one in GF(2) other in R

In summary, H is a nxn matrix with elements in {0,1} and G is a nxn matrix with elements in GF(2). The vector Gm acts as a "choice function," selecting which columns of H to be summed in the output, which will be a real-valued vector. Multiplying two matrices, one with elements in GF(2) and the other in R, can be done by combining them into one transformation, denoted as P. However, (HG)m and H(Gm) may not yield the same results in general.
  • #1
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Homework Statement


H is a nxn matrix with elements in {0,1}
G is a nxn matrix with elements in GF(2)
m is a nx1 vector with elements in GF(2).
How can we perceive the output of
HGm where Gm multiplication is in GF(2) and H multiplication is a normal real multiplication.
Actually I want to combine HG transformation into one P transformation. How can I multiply two matrices while elements in one is in GF(2) and other is in R ?
(We can also restrict the entries in H to be one of 0 and 1 but the output can be in R).


Homework Equations





The Attempt at a Solution


 
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  • #2
the entries in the vector Gm will still be in GF(2) = {0,1}.

so basically the vector Gm acts as a "choice function" picking out which columns of H get summed in the ouput, which will be a real-valued vector.

it makes more sense to do it this way, than to imagine what "HG" means (in this case G acts in a more complicated way, which is then subjected to another selection via m).

in general, (HG)m and H(Gm) won't yield the same results:

[a b]([1 0][1])
[c d]([1 1][1]) =

[a b][1]
[c d][0], which is (a,c)

([a b][1 0])[1]
([c d][1 1])[1] =

[a+b b][1]
[c+d d][1] = (a+2b,c+2d), which is only equal to the first mod 2.
 

1. What is the purpose of multiplying two matrices, one in GF(2) and the other in R?

The purpose of multiplying two matrices, one in GF(2) and the other in R, is to perform a mathematical operation that combines the elements of each matrix in a specific way. This can be useful in various applications, such as solving systems of linear equations and representing transformations in computer graphics.

2. How is the multiplication of two matrices, one in GF(2) and the other in R, different from traditional matrix multiplication?

The multiplication of two matrices, one in GF(2) and the other in R, is different from traditional matrix multiplication in that the elements are multiplied using the rules of modular arithmetic. In GF(2), the only possible elements are 0 and 1, and addition and multiplication are performed modulo 2. In R, the elements can be any real numbers and the standard rules of matrix multiplication apply.

3. Can the multiplication of two matrices, one in GF(2) and the other in R, result in a matrix with elements outside of GF(2) or R?

No, the multiplication of two matrices, one in GF(2) and the other in R, will always result in a matrix with elements within GF(2) or R. In GF(2), the elements are limited to 0 and 1, and in R, the elements are limited to real numbers. Therefore, the product of the two matrices will also have elements within these sets.

4. What is the significance of using GF(2) in the multiplication of matrices?

The use of GF(2) in the multiplication of matrices is significant because it allows for efficient computation and representation of binary systems. GF(2) is commonly used in coding theory and cryptography, where operations need to be performed on binary data. Multiplication in GF(2) also has applications in error-correcting codes and coding theory.

5. Is the multiplication of two matrices, one in GF(2) and the other in R, commutative?

No, the multiplication of two matrices, one in GF(2) and the other in R, is not commutative. This means that changing the order of the matrices in the multiplication will result in a different product. In other words, AB is not necessarily equal to BA, where A and B are matrices in GF(2) and R, respectively.

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