# Relating a lower triangular Toeplitz matrix to a diagonal matrix

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?

• I suspect that it is not possible to write D=ATB or T=ADB since the determinant of D is x1*x2*...xn, and the determinant of T is x1*x1*...x1 which implies the determinant of AB is either (x1*x2*...xn)/(x1^n) or (x1^n)/(x1*x2*...xn). This suggests that A and/or B must have x's as entries. – InquisitivePerson Oct 20 '16 at 23:28