Hadamard product to Matrix product transformation

In summary, the conversation discusses the search for a function that maps between two matrices, A and B, with specific ranges and properties. The conversation also touches on the relationship between element-wise matrix multiplication and regular matrix multiplication. While some potential solutions are suggested, a counter example is presented that disproves the original formula. The conversation ends with a request for any additional information or references on the topic.
  • #1
Xadan
2
0
I've been trying to find an answer to this for some time now, if anyone can provide a definitive answer, they'll have the satisfaction of knowing that several University mathematics professors haven't been able to come up with one...

I am looking for a way to transform from a Hadamard product (element-wise matrix multiplication) into a classic matrix product. Before I get flamed for posing a supposedly stupid question, the answer is related to quantum networks - (see -

http://archive.numdam.org/ARCHIVE/AIF/AIF_1999__49_3/AIF_1999__49_3_927_0/AIF_1999__49_3_927_0.pdf" [Broken]

p. 951).


So more specifically - Let us start with a matrix Z, Z is an element of Real. Element wise multiplication is given by .*, regular multiplication is * . A ranges between 0 and positive infinity, B ranges between -infinity and positive infinity.

A.*Z = B*Z

Is there a function that maps A to B?


To provide more information, think about B. If we exponentiate it to 0, we get the identity matrix (I) . If we exponentiate it by -1 (assuming it does have an inverse) then we get Z. On the left hand side of the equation - if A is uniform (i.e. all matrix elements are identical), then we get Z. If A is the identity matrix, then we get (I). I'm chiefly interested in what happens in the inbetween states (i.e. between no change and orthogonalization). Transitioning B is straightforward (exponentiate to p where p ranges between 0 and -1). Transitioning A is less so. I get the feeling that the answer has something to do with rotation, but it's n-dimensional...

Any advice (or references) would be appreciated. Thanks!
 
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  • #2
To map between A and B, I guess you can use A = (BZ)./Z, and B = (A.*Z)Z^-1. The function is probably dependent on the structure of Z, and therefore Z can't be separated out.

If that's the case, then what is the function that transitions from identity (I) through Z and then to uniform (all values are 1/n)?
 
  • #3
Hi,

I was wondering if you found an answer to your question. I stumbled on a similar problem.

Thanks.
 
  • #4
The formula A.*Z=B*Z is not possible in general. Here is a counter example using a matrix rank inequality.

We have rank(A.*Z)=rank(B*Z)<=min(rank(B),rank(Z)).

Let A=[1 -1;1 1]' and Z=[1 1;1 1]', then rank(A)=2 and rank(Z)=1. Pick whatever B you like,

then 2=rank(A)=rank(A.*Z)<=min(rank(B),rank(Z))=1 which is impossible

and we have a counter example.
 
  • #5


The transformation from Hadamard product to matrix product is a well-studied topic in mathematics, often referred to as the Kronecker product. This transformation is commonly used in quantum mechanics, signal processing, and linear algebra. The Kronecker product is defined as the tensor product of two matrices, which is essentially a way to combine two matrices into a larger, block matrix.

To transform from Hadamard product to matrix product, you can use the Kronecker product as follows:

If A and B are two matrices of dimensions m x n and p x q respectively, then their Kronecker product A ⊗ B is a matrix of dimensions mp x nq given by:

A ⊗ B = [a11B a12B ... a1nB; a21B a22B ... a2nB; ... ; am1B am2B ... amnB]

where each entry aij is multiplied by the entire B matrix.

In your specific case, if A.*Z = B*Z, then you can transform this into matrix form as:

(A ⊗ I)*Z = (B ⊗ I)*Z

where I is the identity matrix of size m x m. This transformation allows you to use regular matrix multiplication to solve for Z.

I hope this helps with your question and provides some clarity on the relationship between Hadamard and matrix products.
 

1. What is a Hadamard product?

A Hadamard product is a mathematical operation that takes two matrices with the same dimensions and multiplies each element in the first matrix by the corresponding element in the second matrix. The result is a new matrix with the same dimensions as the original matrices.

2. How is the Hadamard product different from the standard matrix product?

The Hadamard product operates element-wise, meaning that it multiplies each element in one matrix by the corresponding element in the other matrix. In contrast, the standard matrix product multiplies entire rows and columns of matrices to produce a new matrix. This means that the dimensions of the matrices involved in the Hadamard product must be the same, while the dimensions of matrices involved in standard matrix product must be compatible (e.g. a 3x2 matrix can be multiplied by a 2x4 matrix).

3. What are the benefits of using the Hadamard product?

The Hadamard product is useful in many areas of mathematics and science, but it is particularly beneficial in matrix operations involving data sets. It allows for efficient and simple element-wise operations on matrices, making it a popular choice for manipulating and analyzing large data sets.

4. How is the Hadamard product used in the transformation from Hadamard product to matrix product?

In order to transform a Hadamard product into a matrix product, the Hadamard product must be converted into a diagonal matrix. This is done by taking the diagonal elements of the Hadamard product and placing them on the diagonal of a new matrix, with all other elements being equal to zero. The resulting diagonal matrix can then be used in the matrix product to achieve the desired transformation.

5. Are there any limitations to using the Hadamard product in matrix operations?

One potential limitation of using the Hadamard product is that it can only be used on matrices with the same dimensions. This means that it cannot be used in cases where the matrices have different dimensions, as is often the case in more complex mathematical operations. Additionally, the Hadamard product does not take into account the relationship between different elements in a matrix, making it less suitable for certain types of data analysis.

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