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I found following two definitions in a Ben Israel's book whose title is Generalized inverses: Theory and applications:

For any $A, B \in \mathbb{C}^{m\times n}$, define

$R (A, B) = \{Y = A X B \in \mathbb{C}^{m\times n}: X \in \mathbb{C}^{n\times m} \}$

and

$N (A, B) = \{X \in \mathbb{C}^{n\times m} : A X B = 0 \}$

which we shall call the range and null space of $(A, B)$, respectively. I am confused with this definition. In what context we are calling this range and null spaces of (A, B)? Is there any reference that might help me?

Could any body help me to understand this definition?

thanks

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$X \mapsto AXB$ is a linear transformation, is it not? –  Hurkyl Oct 5 '12 at 9:35
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1 Answer

To me this looks like the superoperator formalism. Think of your product matrices as vectors, then the following holds: $$ \text{vec}(AXB) = (B^T \otimes A) \text{vec}(X). $$ (see here for a definition of $\text{vec}(X)\;$ and here for more information: Kronecker product)

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