# Factoring out vector parameter from block diagonal matrix

I have a block diagonal matrix $$\mathbf{A}$$ of size $$NK\times K$$, whose elements are copies of a column vector $$\mathbf{a}=[a_1, \dots, a_n]^T$$ with $$n=1,\dots,N$$ and of size $$N \times 1$$. Such a block diagonal matrix reads as follows:

$$\mathbf{A}=\begin{bmatrix} \mathbf{a} & 0 & \dots & 0\\ 0 & \mathbf{a} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \dots & \mathbf{a} \\ \end{bmatrix}= \begin{bmatrix} a_1 & 0 & \dots & 0\\ a_2 & 0 & \dots & 0\\ \vdots & 0 & \dots & 0\\ a_N & 0 & \dots & 0\\ 0 & a_1 & \dots & 0\\ 0 & a_2 & \dots & 0\\ 0 & \vdots & \dots & 0\\ 0 & a_N & \dots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \dots & a_1 \\ 0 & 0 & \dots & a_2 \\ 0 & 0 & \dots & \vdots \\ 0 & 0 & \dots & a_N \\ \end{bmatrix}.$$

Is it possible to factor out the vector $$\mathbf{a}$$, or $$\mathbf{a}^T$$ from $$\mathbf{A}$$ ? I would like to have either $$\mathbf{a}$$, or $$\mathbf{a}^T$$ as the rigthmost or leftmost term.

• Just looking at the dimensions should give you a hint - pre multiply by a $N \times NK$ matrix and post multiply by a $K \times 1$ matrix. I leave it to you to figure out what the entries should be Jun 30, 2021 at 10:56

If I understand the question correctly, you would like to write $$\mathbf A = \mathbf a \cdot {?}$$, $$\mathbf A = \mathbf a^T \cdot {?}$$, $$\mathbf A = {?} \cdot \mathbf a$$ or $$\mathbf A = {?} \cdot \mathbf a^T$$.
Since $$\mathbf A$$ is a $$NK\times K$$ matrix and $$\mathbf a$$ is $$N\times 1$$, neither of these factorizations can work with a usual matrix product, where “$$(k\times l)\cdot(l\times m)=k\times m$$”.
However, your matrix $$\mathbf A$$ can be written as a Kronecker product: $$\mathbf A = \mathbf I_k \otimes \mathbf a.$$ Here $$\mathbf I_k$$ denotes the $$k\times k$$ identity matrix.