Condition to make a matrix symmetric and feasibility of the solution

Question

Given a diagonal matrix $\mathbf{D}$ of size $R^{n \times n}$ and a full rank matrix $\mathbf{B}$ of size $R^{n \times n}$, what would be the conditions on $\mathbf{B}$ such that $\mathbf{DB}$ is symmetric? and how feasible is the solution?

Edit 1: My first attempt was to decompose $\mathbf{B} = \mathbf{US_1V^\top}$ using SVD such that the following holds true

\begin{align*} \mathbf{D}\mathbf{USV^\top} = \mathbf{S_2}\mathbf{E}\mathbf{S_2^\top}, \end{align*} where $\mathbf{S_2}\mathbf{E}\mathbf{S_2^\top}$ is SVD of $\mathbf{D}\mathbf{USV^\top}$ and then use the above equation to check if that thing holds true or not. Is it just sufficient to find constraints on $\mathbf{B}$ such that $\mathbf{DB}$ is symmetric or should there be conditions on $\mathbf{D}$ as well?

Answer 1

By definition of matrix multiplication and diagonal, \begin{align} (DB)_{ij} &= \sum_{k=1}^n D_{ik} B_{kj} = D_{ii} B_{ij} \\ (DB)_{ji} &= \sum_{k=1}^n D_{jk} B_{ki} = D_{jj} B_{ji} \end{align} Symmetric means these two are equal for all $i$ and $j$.