Kronecker Product: Solving AA` with w & D

In summary, The question posed is whether the matrix AA` can be written as the kronecker product of two matrices, given that A is a symmetric matrix and B is a n x n matrix. It is stated that every matrix B can be written as a Kronecker product of X\otimes Y, where X=B and Y is 1x1 identity matrix. However, the desired multiplication (A kronecker product B) by (AA`) is not possible due to the dimensions of AA` remaining unchanged. The asker is seeking further ideas or solutions.
  • #1
Shurid
2
0
Hi everyone,

Please help me with this problem.

Suppose w be a n x n symmetric matrix and D be n x m matrix.

Let A=wDD`w.

Is it possible to write the matrix, AA`= (wDD`w)(wDD`w) as the kronecker product of any two matrices?

Thanks in advance.
 
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  • #2
Every matrix B is a Kronecker product [itex]X\otimes Y[/itex], where X=B and Y is 1x1 identity matrix.
 
  • #3
Thanks a lot Arkazad. You are right. However, what I want is to multiply

(A kronecker product B) by (AA`), where B is also a n x n matrix. You observe that this is not even defined. I don't know may be this is impossible, because I can not change the dimension of AA`.

Any further idea?
 

FAQ: Kronecker Product: Solving AA` with w & D

What is the Kronecker product?

The Kronecker product, denoted as ⊗, is a mathematical operation that combines two matrices to form a larger matrix. It is named after German mathematician Leopold Kronecker and is often used in linear algebra and signal processing.

How do you solve AA` with w and D using the Kronecker product?

To solve AA` with w and D, you can use the following equation: (A ⊗ I)vec(w) = vec(D), where A is a matrix, I is the identity matrix, w is a vector, and D is a matrix. This equation can be solved using matrix algebra to find the vector w.

3. What is the significance of using the Kronecker product in solving AA` with w and D?

The Kronecker product allows us to represent a large matrix equation in a more compact form. This makes it easier to solve, especially for systems with a large number of variables. The Kronecker product also has many applications in mathematics, physics, and engineering.

4. Can the Kronecker product be used in other types of equations?

Yes, the Kronecker product can be used in a variety of equations, such as in eigenvalue problems, system of linear equations, and Markov processes. It is a versatile tool in linear algebra and can be applied in many different contexts.

5. What are some limitations of using the Kronecker product to solve AA` with w and D?

The Kronecker product can become computationally expensive for large matrices, making it impractical for some applications. It also assumes that the matrices being multiplied are compatible, which may not always be the case. Additionally, the solutions generated by the Kronecker product may not always be unique.

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