1
$\begingroup$

Let $A=[a_{ij}]$ be an $m \times m$ matrix and $B$ be a $nm \times nm$ block diagonal matrix with diagonal blocks $B_i$, $i=1, \ldots, m$, $n \times n$ matrices. I want to express

$\left[ \begin{array}[cccc] \\a_{11} B_1 & a_{12}B_1 & \cdots & a_{1m} B_1 \\ a_{21} B_2 & a_{22}B_2 & \cdots & a_{2m} B_2 \\ \vdots & \vdots & \vdots & \vdots\\ a_{m1} B_m & a_{m2}B_m & \cdots & a_{mm} B_m \\ \end{array}\right]_{mn \times mn}$

in a compact form, possibly in one shot using Kronecker products. Is it possible?

$\endgroup$
1
$\begingroup$

Let $e_k$ be the $k$-th column vector of the standard basis of $\mathbb{R}^m$ and $I_n$ the $n \times n$ identity matrix. Thus we can write $$ \sum_{k=1}^{m} (e_k \otimes I_n)((e_k^T A) \otimes B_k) \tag{1} $$

Lets see how we construct this expression.

The term $(e_k^T A)$ is just the $k$ row of $A$ $$ (e_k^T A) = \left[ \begin{matrix} a_{k1} & \cdots & a_{km} \end{matrix} \right]_{1 \times m} $$

Hence $$ (e_k^T A) \otimes B_k = \left[ \begin{matrix} a_{k1}B_k & \cdots & a_{km}B_k \end{matrix} \right]_{n \times mn} $$

Then we use $$ (e_k \otimes I_n) = \left[ \begin{matrix} 0 \\ \vdots \\ I_n \\ \vdots \\ 0 \end{matrix} \right]_{mn \times n} $$

to "immerse" $(e_k^T A) \otimes B_k$ as the $k$-th row of a $mn \times mn$ matrix $$ (e_k \otimes I_n)((e_k^T A) \otimes B_k) = \left[ \begin{matrix} 0 & \cdots & 0 \\ \vdots & \vdots & \vdots \\ a_{k1} B_k & \cdots & a_{km} B_k \\ \vdots & \vdots & \vdots \\ 0 & \cdots & 0 \\ \end{matrix} \right]_{mn \times mn} $$

And then finnaly we add up all these $m$ matrices.

Now we can use the Kronecker product "mixed-product" property to simplify $(1)$ as $$ \sum_{k=1}^{m} (e_k \otimes I_n)((e_k^T A) \otimes B_k) = \sum_{k=1}^{m} ((e_k e_k^T A) \otimes (I_n B_k)) = \sum_{k=1}^{m} (e_k e_k^T A) \otimes B_k \tag{2} $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.