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We have the following matrix

$\left[\begin{array}{c} A\otimes B_1\\ \vdots\\ A\otimes B_T \end{array}\right]$, where $A$ and the $B_i$ are matrices, and $\otimes$ is the Kronecker product.

Is it possible to rewrite it as a Kronecker product where we gather all the $B_i$ matrices as one big matrix?

For example something like $(\cdots) \otimes \left[\begin{array}{c} B_1\\ \vdots\\ B_T \end{array}\right]$

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You could use the fact that $$\mathbf A \otimes \mathbf B = \mathbf K_{M,N} \cdot \left( \mathbf B \otimes \mathbf A\right) \cdot \mathbf K_{P,Q}^{\rm T},$$ for an $\mathbf A \in \mathbb{F}^{M \times P}$, $\mathbf B \in \mathbb{F}^{N \times Q}$ and $\mathbb{F}$ is the field you are considering.

Here, $\mathbf K_{M,N}$ represents the commutation matrix of size $MN \times MN$, which is the permutation matrix defined via $$\mathbf K_{M,N}\cdot {\rm vec}\{\mathbf X\} = {\rm vec}\{\mathbf X^{\rm T}\}$$ for any $\mathbf X \in \mathbb{F}^{M \times N}$. A lot is known on these and they have some fun properties. Magnus and Neudecker [1,2] would be a good source to study.

You could apply this to each row to "turn around" all your Kronecker products, which then allows to pull out $\mathbf A$ completely. This should give you something like a ${\rm diag}\{\mathbf K_1, \ldots, \mathbf K_T\}$ in front, which may be a bit unhandy. But at least you know that such permutation matrices exist.

Not sure if this helps anything!

[1] Magnus, Jan R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Stat. 7, 381-394 (1979). ZBL0414.62040.

[2] Magnus, Jan R.; Neudecker, Heinz, Matrix differential calculus with applications in statistics and econometrics, Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. Chichester etc.: John Wiley & Sons. XVII, 393 p.; £ 24.50 (1988). ZBL0651.15001.

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