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## Main Question or Discussion Point

Hi there,

I was recently working with Kronecker product of matrices, and a question came up that I'm not sure how to answer. Is the matrix that represents a Kronecker product of two infinite dimensional matrices well defined? If yes, are some of the properties of the Kronecker product listed in http://en.wikipedia.org/wiki/Kronecker_product satisfied for the infinite dimensional matrices:

1. [itex]A \otimes (B + C) = A \otimes B + A \otimes C [/itex]

2. [itex]k (A \otimes B) = (kA \otimes B)[/itex]

3. [itex](A \otimes B)(C \otimes D) = AC \otimes BD[/itex]

On one hand, it seems tricky to define such a product matrix because it is supposed to look like a block matrix, but how can it exist if each block is going to be of an infinite size?

On the other hand, looking on the literature about the tensor product of linear operators, it seems possible to define a product linear operator, even if the vector space that these operators work on are infinite dimensional. In this case, if I'm not mistaken, infinite dimensional matrices are instances of these linear operations.

Any clarifications are much appreciated!

Yuri

I was recently working with Kronecker product of matrices, and a question came up that I'm not sure how to answer. Is the matrix that represents a Kronecker product of two infinite dimensional matrices well defined? If yes, are some of the properties of the Kronecker product listed in http://en.wikipedia.org/wiki/Kronecker_product satisfied for the infinite dimensional matrices:

1. [itex]A \otimes (B + C) = A \otimes B + A \otimes C [/itex]

2. [itex]k (A \otimes B) = (kA \otimes B)[/itex]

3. [itex](A \otimes B)(C \otimes D) = AC \otimes BD[/itex]

On one hand, it seems tricky to define such a product matrix because it is supposed to look like a block matrix, but how can it exist if each block is going to be of an infinite size?

On the other hand, looking on the literature about the tensor product of linear operators, it seems possible to define a product linear operator, even if the vector space that these operators work on are infinite dimensional. In this case, if I'm not mistaken, infinite dimensional matrices are instances of these linear operations.

Any clarifications are much appreciated!

Yuri