Suppose I had a lower triangular Toeplitz matrix:
$$ T= \begin{bmatrix} x_{1} & 0 & 0 & \dots & 0 \\ x_{2} & x_{1} & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{n} & x_{n-1} & x_{n-2} & \dots & x_{1} \end{bmatrix} $$ and a diagonal matrix: $$ D= \begin{bmatrix} x_{1} & 0 & 0 & \dots & 0 \\ 0 & x_{2} & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & x_{n} \end{bmatrix} $$
where the diagonal elements are the elements of $T$.
Is it possible to relate $T$ and $D$ in either of the following forms:
$D=ATB$
$T=ADB$
where $A$ and $B$ are square matrices that do not have $x$'s as entries? If not, is there another way to relate them using matrix equations?