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I am reading some notes about tensor product of vector spaces (those in here) where the following sentence can be found:

Note that not every matrix on V ⊗ W can be written as a tensor product of a matrix on V and another on W.

Matrices in V ⊗ W are defined as a Kronecker product. So, how could this claim be possible? Has it to do with the problem of splitting the Kronecker product of two matrices?

More generally: does the general picture changes if we consider V and W infinite-dimensional vector spaces?

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Two elements:

  1. If you read the sentence following the one you quote, there is an explanation based on dimensions.
  2. Consider a $4 \times 4$ matrix $C$ with only one zero entry. If you try to write $C = A \otimes B$, either none of the $A,B$ entries is equal to zero and then $C$ will have no zero entry. Or they have at least one, and then $C$ will have at least four zero entries.

The second argument can be extended for infinite dimensions.

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  • $\begingroup$ Thank you for your clarity. Does this fact mean that not every basis of V⊗W is of the form of a tensor product of a basis of V and a basis of W? $\endgroup$ – gibarian Jun 2 at 13:33
  • $\begingroup$ You can refer here for some elements. $\endgroup$ – mathcounterexamples.net Jun 2 at 13:38

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