Assume there is a vector, $$ A = \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_m \end{bmatrix} \in \mathbb R^{mn}, $$ and each subvector $a_i, i\in \{1,2,\cdots,m\}$ are $n$-by-$1$ vector.
The output is $$ B = \begin{bmatrix} a_1^T & 0 & \cdots & 0 & 0 \\ 0 & a_2^T & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & a_{m-1}^T & 0 \\ 0 & 0 & 0 & 0 & a_m^T \end{bmatrix}\in \mathbb R^{m\times mn}. $$
How to use matrix operations to obtain $B$ from $A$?
(Here, we exclude the method of defining a mapping from $A$ to $B$. )
The transformation can be described by $$B = (I_m\otimes{\tt1}_n^T)\cdot\operatorname{Diag}(A)$$ where $I_m$ is the $m\times m$ identity matrix, ${\tt1}_n$ is the $n\times 1$ all-ones vector, $\otimes$ denotes the Kronecker product, and the Diag operation creates a diagonal matrix from its vector argument.
If Diag is deemed an unacceptable operation, it can be replaced with a Hadamard $(\odot)$ product. $${\rm Diag}(a) = I_{mn}\odot (a{\tt1}_{mn}^T)$$